BEHAVIORAL AND BRAIN SCIENCES (2013) 36, 255–327
doi:10.1017/S0140525X12001525
Can quantum probability provide a
new direction for cognitive modeling?
Emmanuel M. Pothos
Department of Psychology, City University London, London EC1V 0HB,
United Kingdom
[email protected]
http://www.staff.city.ac.uk/∼sbbh932/
Jerome R. Busemeyer
Department of Psychological and Brain Sciences, Indiana University,
Bloomington, IN 47405
[email protected]
http://mypage.iu.edu/∼jbusemey/home.html
Abstract: Classical (Bayesian) probability (CP) theory has led to an influential research tradition for modeling cognitive processes.
Cognitive scientists have been trained to work with CP principles for so long that it is hard even to imagine alternative ways to
formalize probabilities. However, in physics, quantum probability (QP) theory has been the dominant probabilistic approach for
nearly 100 years. Could QP theory provide us with any advantages in cognitive modeling as well? Note first that both CP and QP
theory share the fundamental assumption that it is possible to model cognition on the basis of formal, probabilistic principles. But
why consider a QP approach? The answers are that (1) there are many well-established empirical findings (e.g., from the influential
Tversky, Kahneman research tradition) that are hard to reconcile with CP principles; and (2) these same findings have natural and
straightforward explanations with quantum principles. In QP theory, probabilistic assessment is often strongly context- and orderdependent, individual states can be superposition states (that are impossible to associate with specific values), and composite systems
can be entangled (they cannot be decomposed into their subsystems). All these characteristics appear perplexing from a classical
perspective. However, our thesis is that they provide a more accurate and powerful account of certain cognitive processes. We first
introduce QP theory and illustrate its application with psychological examples. We then review empirical findings that motivate the
use of quantum theory in cognitive theory, but also discuss ways in which QP and CP theories converge. Finally, we consider the
implications of a QP theory approach to cognition for human rationality.
Keywords: category membership; classical probability theory; conjunction effect; decision making; disjunction effect; interference
effects; judgment; quantum probability theory; rationality; similarity ratings
1. Preliminary issues
1.1. Why move toward quantum probability theory?
In this article we evaluate the potential of quantum probability (QP) theory for modeling cognitive processes.
What is the motivation for employing QP theory in cognitive modeling? Does the use of QP theory offer the
promise of any unique insights or predictions regarding
cognition? Also, what do quantum models imply regarding
the nature of human rationality? In other words, is there
anything to be gained, by seeking to develop cognitive
models based on QP theory? Especially over the last
decade, there has been growing interest in such models,
encompassing publications in major journals, special
issues, dedicated workshops, and a comprehensive book
(Busemeyer & Bruza 2012). Our strategy in this article is
to briefly introduce QP theory, summarize progress with
selected, QP models, and motivate answers to the abovementioned questions. We note that this article is not
about the application of quantum physics to brain physiology. This is a controversial issue (Hammeroff 2007; Litt
et al. 2006) about which we are agnostic. Rather, we are
interested in QP theory as a mathematical framework for
© Cambridge University Press 2013
0140-525X/13 $40.00
cognitive modeling. QP theory is potentially relevant in
any behavioral situation that involves uncertainty. For
example, Moore (2002) reported that the likelihood of a
“yes” response to the questions “Is Gore honest?” and “Is
Clinton honest?” depends on the relative order of the questions. We will subsequently discuss how QP principles can
provide a simple and intuitive account for this and a range
of other findings.
QP theory is a formal framework for assigning probabilities to events (Hughes 1989; Isham 1989). QP theory can
be distinguished from quantum mechanics, the latter being
a theory of physical phenomena. For the present purposes,
it is sufficient to consider QP theory as the abstract foundation of quantum mechanics not specifically tied to
physics (for more refined characterizations see, e.g., Aerts
& Gabora 2005b; Atmanspacher et al. 2002; Khrennikov
2010; Redei & Summers 2007). The development of
quantum theory has been the result of intense effort
from some of the greatest scientists of all time, over a
period of >30 years. The idea of “quantum” was first proposed by Planck in the early 1900s and advanced by Einstein. Contributions from Bohr, Born, Heisenberg, and
Schrödinger all led to the eventual formalization of QP
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theory by von Neumann and Dirac in the 1930s. Part of the
appeal of using QP theory in cognition relates to confidence
in the robustness of its mathematics. Few other theoretical
frameworks in any science have been scrutinized so intensely, led to such surprising predictions, and, also, changed
human existence as much as QP theory (when applied to
the physical world; quantum mechanics has enabled the
development of, e.g., the transistor, and, therefore, the
microchip and the laser).
QP theory is, in principle, applicable not just in physics,
but in any science in which there is a need to formalize
uncertainty. For example, researchers have been pursuing
applications in areas as diverse as economics (Baaquie
2004) and information theory (e.g., Grover 1997; Nielsen
& Chuang 2000). The idea of using quantum theory in psychology has existed for nearly 100 years: Bohr, one of the
founding fathers of quantum theory, was known to
believe that aspects of quantum theory could provide
insight about cognitive process (Wang et al., in press).
However, Bohr never made any attempt to provide a
formal cognitive model based on QP theory, and such
models have started appearing only fairly recently (Aerts
EMMANUEL POTHOS studied physics at Imperial
College, during which time he obtained the Stanley
Raimes Memorial prize in mathematics, and continued
with a doctorate in experimental psychology at Oxford
University. He has worked with a range of computational frameworks for cognitive modeling, including
ones based on information theory, flexible representation spaces, Bayesian methods, and, more recently,
quantum theory. He has authored approximately sixty
journal articles on related topics, as well as on applications of cognitive methods to health and clinical psychology. Pothos is currently a senior lecturer in
psychology at City University London.
JEROME BUSEMEYER received his PhD as a mathematical psychologist from University of South Carolina in
1980, and later he enjoyed a post-doctoral position at
University of Illinois. For 14 years he was a faculty
member at Purdue University. He moved on to
Indiana University, where he is provost professor, in
1997. Busemeyer’s research has been steadily funded
by the National Science Foundation, National Institute
of Mental Health, and National Institute on Drug
Abuse, and in return he served on national grant
review panels for these agencies. He has published
over 100 articles in various cognitive and decision
science journals, such as Psychological Review, as well
as serving on their editorial boards. He served as chief
editor of Journal of Mathematical Psychology from
2005 through 2010 and he is currently an associate
editor of Psychological Review. From 2005 through
2007, Busemeyer served as the manager of the Cognition and Decision Program at the Air Force Office of
Scientific Research. He became a fellow of the Society
of Experimental Psychologists in 2006. His research
includes mathematical models of learning and decision
making, and he formulated a dynamic theory of
human decision making called decision field theory.
Currently, he is working on a new theory applying
quantum probability to human judgment and decision
making, and he published a new book on this topic
with Cambridge University Press.
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& Aerts 1995; Aerts & Gabora 2005b; Atmanspacher
et al. 2004; Blutner 2009; Bordley 1998; Bruza et al.
2009; Busemeyer et al. 2006b; Busemeyer et al. 2011;
Conte et al. 2009; Khrennikov 2010; Lambert-Mogiliansky
et al. 2009; Pothos & Busemeyer 2009; Yukalov & Sornette
2010). But what are the features of quantum theory that
make it a promising framework for understanding cognition? It seems essential to address this question before
expecting readers to invest the time for understanding
the (relatively) new mathematics of QP theory.
Superposition, entanglement, incompatibility, and interference are all related aspects of QP theory, which endow
it with a unique character. Consider a cognitive system,
which concerns the cognitive representation of some information about the world (e.g., the story about the hypothetical Linda, used in Tversky and Kahneman’s [1983] famous
experiment; sect. 3.1 in this article). Questions posed to
such systems (“Is Linda feminist?”) can have different outcomes (e.g., “Yes, Linda is feminist”). Superposition has to
do with the nature of uncertainty about question outcomes.
The classical notion of uncertainty concerns our lack of
knowledge about the state of the system that determines
question outcomes. In QP theory, there is a deeper
notion of uncertainty that arises when a cognitive system
is in a superposition among different possible outcomes.
Such a state is not consistent with any single possible
outcome (that this is the case is not obvious; this remarkable
property follows from the Kochen–Specker theorem).
Rather, there is a potentiality (Isham 1989, p. 153) for
different possible outcomes, and if the cognitive system
evolves in time, so does the potentiality for each possibility.
In quantum physics, superposition appears puzzling: what
does it mean for a particle to have a potentiality for different
positions, without it actually existing at any particular position? By contrast, in psychology, superposition appears an
intuitive way to characterize the fuzziness (the conflict,
ambiguity, and ambivalence) of everyday thought.
Entanglement concerns the compositionality of complex
cognitive systems. QP theory allows the specification of
entangled systems for which it is not possible to specify a
joint probability distribution from the probability distributions of the constituent parts. In other words, in entangled
composite systems, a change in one constituent part of the
system necessitates changes in another part. This can lead
to interdependencies among the constituent parts not possible in classical theory, and surprising predictions, especially
when the parts are spatially or temporally separated.
In quantum theory, there is a fundamental distinction
between compatible and incompatible questions for a cognitive system. Note that the terms compatible and incompatible have a specific, technical meaning in QP theory, which
should not be confused with their lay use in language. If
two questions, A and B, about a system are compatible, it
is always possible to define the conjunction between A
and B. In classical systems, it is assumed by default that
all questions are compatible. Therefore, for example, the
conjunctive question “are A and B true” always has a yes
or no answer and the order between questions A and B
in the conjunction does not matter. By contrast, in QP
theory, if two questions A and B are incompatible, it is
impossible to define a single question regarding their conjunction. This is because an answer to question A implies a
superposition state regarding question B (e.g., if A is true at
a time point, then B can be neither true nor false at the
Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
same time point). Instead, QP defines conjunction between
incompatible questions in a sequential way, such as “A and
then B.” Crucially, the outcome of question A can affect the
consideration of question B, so that interference and order
effects can arise. This is a novel way to think of probability,
and one that is key to some of the most puzzling predictions
of quantum physics. For example, knowledge of the position of a particle imposes uncertainty on its momentum.
However, incompatibility may make more sense when considering cognitive systems and, in fact, it was first introduced in psychology. The physicist Niels Bohr borrowed
the notion of incompatibility from the work of William
James. For example, answering one attitude question can
interfere with answers to subsequent questions (if they
are incompatible), so that their relative order becomes
important. Human judgment and preference often
display order and context effects, and we shall argue that
in such cases quantum theory provides a natural explanation of cognitive process.
1.2. Why move away from existing formalisms?
By now, we hope we have convinced readers that QP
theory has certain unique properties, whose potential for
cognitive modeling appears, at the very least, intriguing.
For many researchers, the inspiration for applying
quantum theory in cognitive modeling has been the widespread interest in cognitive models based on CP theory
(Anderson 1991; Griffiths et al. 2010; Oaksford & Chater
2007; Tenenbaum et al. 2011). Both CP and QP theories
are formal probabilistic frameworks. They are founded on
different axioms (the Kolmogorov and Dirac/von
Neumann axioms, respectively) and, therefore, often
produce divergent predictions regarding the assignment
of probabilities to events. However, they share profound
commonalities as well, such as the central objective of
quantifying uncertainty, and similar mechanisms for
manipulating probabilities. Regarding cognitive modeling,
quantum and classical theorists share the fundamental
assumption that human cognition is best understood
within a formal probabilistic framework.
As Griffiths et al. (2010, p. 357) note, “probabilistic
models of cognition pursue a top-down or ‘function-first’
strategy, beginning with abstract principles that allow
agents to solve problems posed by the world … and then
attempting to reduce these principles to psychological
and neural processes.” That is, the application of CP
theory to cognition requires a scientist to create hypotheses
regarding cognitive representations and inductive biases
and, therefore, elucidate the fundamental questions of
how and why a cognitive problem is successfully addressed.
In terms of Marr’s (1982) analysis, CP models are typically
aimed at the computational and algorithmic levels,
although perhaps it is more accurate to characterize them
as top down or function first (as Griffiths et al. 2010,
p. 357).
We can recognize the advantage of CP cognitive models
in at least two ways. First, in a CP cognitive model, the principles that are invoked (the axioms of CP theory) work as a
logical “team” and always deductively constrain each other.
By contrast, alternative cognitive modeling approaches
(e.g., based on heuristics) work “alone” and therefore are
more likely to fall foul of arbitrariness problems, whereby
it is possible to manipulate each principle in the model
independently of other principles. Second, neuroscience
methods and computational bottom-up approaches are
typically unable to provide much insight into the fundamental why and how questions of cognitive process (Griffiths et al. 2010). Overall, there are compelling reasons
for seeking to understand the mind with CP theory. The
intention of QP cognitive models is aligned with that of
CP models. Therefore, it makes sense to present QP
theory side by side with CP theory, so that readers can
appreciate their commonalities and differences.
A related key issue is this: if CP theory is so successful
and elegant (at least, in cognitive applications), why seek
an alternative? Moreover, part of the motivation for using
CP theory in cognitive modeling is the strong intuition supporting many CP principles. For example, the probability
of A and B is the same as the probability of B and A
(Prob(A&B)=Prob(A&B)). How can it be possible that
the probability of a conjunction depends upon the order
of the constituents? Indeed, as Laplace (1816, cited in
Perfors et al. 2011) said, “probability theory is nothing
but common sense reduced to calculation.” By contrast,
QP theory is a paradigm notorious for its conceptual difficulties (in the 1960s, Feynman famously said “I think I
can safely say that nobody understands quantum mechanics”). A classical theorist might argue that, when it
comes to modeling psychological intuition, we should
seek to apply a computational framework that is as intuitive
as possible (CP theory) and avoid the one that can lead to
puzzling and, superficially at least, counterintuitive predictions (QP theory).
Human judgment, however, often goes directly against
CP principles. A large body of evidence has accumulated
to this effect, mostly associated with the influential research
program of Tversky and Kahneman (Kahneman et al. 1982;
Tversky & Kahneman 1973; 1974; Tversky & Shafir 1992).
Many of these findings relate to order/context effects, violations of the law of total probability (which is fundamental
to Bayesian modeling), and failures of compositionality.
Therefore, if we are to understand the intuition behind
human judgment in such situations, we have to look for
an alternative probabilistic framework. Quantum theory
was originally developed so as to model analogous effects
in the physical world and therefore, perhaps, it can offer
insight into those aspects of human judgment that seem
paradoxical from a classical perspective. This situation is
entirely analogous to that faced by physicists early in the
last century. On the one hand, there was the strong intuition from classical models (e.g., Newtonian physics, classical electromagnetism). On the other hand, there were
compelling empirical findings that were resisting explanation on the basis of classical formalisms. Therefore, physicists had to turn to quantum theory, and so paved the way
for some of the most impressive scientific achievements.
It is important to note that other cognitive theories
embody order/context effects or interference effects or
other quantum-like components. For example, a central
aspect of the gestalt theory of perception concerns how
the dynamic relationships among the parts of a distal
layout together determine the conscious experience corresponding to the image. Query theory (Johnson et al. 2007)
is a proposal for how value is constructed through a series of
(internal) queries, and has been used to explain the endowment effect in economic choice. In query theory, value is
constructed, rather than read off, and also different
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queries can interfere with each other, so that query order
matters. In configural weight models (e.g., Birnbaum
2008) we also encounter the idea that, in evaluating
gambles, the context of a particular probability-consequence branch (e.g., its rank order) will affect its weight.
The theory also allows weight changes depending upon
the observer perspective (e.g., buyer vs. seller). Anderson’s
(1971) integration theory is a family of models for how a
person integrates information from several sources, and
also incorporates a dependence on order. Fuzzy trace
theory (Reyna 2008; Reyna & Brainerd 1995) is based on
a distinction between verbatim and gist information, the
latter corresponding to the general semantic qualities of
an event. Gist information can be strongly context and
observer dependent and this has led fuzzy trace theory to
some surprising predictions (e.g., Brainerd et al. 2008).
This brief overview shows that there is a diverse range of
cognitive models that include a role for context or order,
and a comprehensive comparison is not practical here.
However, when comparisons have been made, the results
favored quantum theory (e.g., averaging theory was shown
to be inferior to a matched quantum model, Trueblood &
Busemeyer 2011). In some other cases, we can view QP
theory as a way to formalize previously informal conceptualizations (e.g., for query theory and the fuzzy trace theory).
Overall, there is a fair degree of flexibility in the particular specification of computational frameworks in cognitive
modeling. In the case of CP and QP models, this flexibility
is tempered by the requirement of adherence to the axioms
in each theory: all specific models have to be consistent
with these axioms. This is exactly what makes CP (and
QP) models appealing to many theorists and why, as
noted, in seeking to understand the unique features of
QP theory, it is most natural to compare it with CP theory.
In sum, a central aspect of this article is the debate about
whether psychologists should explore the utility of
quantum theory in cognitive theory; or whether the existing
formalisms are (mostly) adequate and a different paradigm
is not necessary. Note that we do not develop an argument
that CP theory is unsuitable for cognitive modeling; it
clearly is, in many cases. And, moreover, as will be discussed, CP and QP processes sometimes converge in
their predictions. Rather, what is at stake is whether
there are situations in which the distinctive features of
QP theory provide a more accurate and elegant explanation
for empirical data. In the next section we provide a brief
consideration of the basic mechanisms in QP theory.
Perhaps contrary to common expectation, the relevant
mathematics is simple and mostly based on geometry and
linear algebra. We next consider empirical results that
appear puzzling from the perspective of CP theory, but
can naturally be accommodated within QP models.
Finally, we discuss the implications of QP theory for understanding rationality.
2. Basic assumptions in QP theory and
psychological motivation
2.1. The outcome space
CP theory is a set-theoretic way to assign probabilities to
the possible outcomes of a question. First, a sample
space is defined, in which specific outcomes about a question are subsets of this sample space. Then, a probability
measure is postulated, which assigns probabilities to disjoint outcomes in an additive manner (Kolmogorov 1933/
1950). The formulation is different in QP theory, which is
a geometric theory of assigning probabilities to outcomes
(Isham 1989). A vector space (called a Hilbert space) is
defined, in which possible outcomes are represented as
subspaces of this vector space. Note that our use of the
terms questions and outcomes are meant to imply the technical QP terms observables and propositions.
A vector space represents all possible outcomes for questions we could ask about a system of interest. For example,
consider a hypothetical person and the general question of
that person’s emotional state. Then, one-dimensional subspaces (called rays) in the vector space would correspond
to the most elementary emotions possible. The number
of unique elementary emotions and their relation to each
other determine the overall dimensionality of the vector
space. Also, more general emotions, such as happiness,
would be represented by subspaces of higher dimensionality. In Figure 1a, we consider the question of whether a
Figure 1. An illustration of basic processes in QP theory. In Figure 1b, all vectors are co-planar, and the figure is a two-dimensional one.
In Figure 1c, the three vectors “Happy, employed,” “Happy, unemployed,” and “Unhappy, employed” are all orthogonal to each other, so
that the figure is a three-dimensional one. (The fourth dimension, “unhappy, unemployed” is not shown).
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hypothetical person is happy or not. However, because it is
hard to picture high multidimensional subspaces, for practical reasons we assume that the outcomes of the happiness
question are one-dimensional subspaces. Therefore, one
ray corresponds to the person definitely being happy and
another one to that person definitely being unhappy.
Our initial knowledge of the hypothetical person is indicated by the state vector, a unit length vector, denoted as
|Ψ〉 (the bracket notation for a vector is called the Dirac
notation). In psychological applications, it often refers to
the state of mind, perhaps after reading some instructions
for a psychological task. More formally, the state vector
embodies all our current knowledge of the cognitive
system under consideration. Using the simple vector space
in Figure 1a, we can write |Ψ〉 = a|happy〉 + b|unhappy〉.
Any vector |Ψ〉 can be expressed as a linear combination of
the |happy〉 and |unhappy〉 vectors, so that these two
vectors form a basis for the two-dimensional space we
have employed. The a and b constants are called amplitudes
and they reflect the components of the state vector along the
different basis vectors.
To determine the probability of the answer happy, we need
to project the state represented by |Ψ〉 onto the subspace for
“happy” spanned by the vector |happy〉. This is done using
what is called a projector, which takes the vector |Ψ〉 and
lays it down on the subspace spanned by |happy〉; this projector can be denoted as Phappy. The projection to the |happy〉
subspace is denoted by Phappy |Ψ〉=a |happy〉. (Here and
elsewhere we will slightly elaborate on some of the basic
definitions in the Appendix.) Then, the probability that
the person is happy is equal to the squared length of the
projection, ||Phappy |Ψ〉||2. That is, the probability that the
person has a particular property depends upon the projection of |Ψ〉 onto the subspace corresponding to the property. In our simple example, this probability reduces to
||Phappy |Ψ〉||2 = |a|2, which is the squared magnitude of
the amplitude of the state vector along the |happy〉 basis
vector. The idea that projection can be employed in psychology to model the match between representations has
been explored before (Sloman 1993), and the QP cognitive
program can be seen as a way to generalize these early
ideas. Also, note that a remarkable mathematical result,
Gleason’s theorem, shows that the QP way for assigning
probabilities to subspaces is unique (e.g., Isham 1989,
p. 210). It is not possible to devise another scheme for
assigning numbers to subspaces that satisfy the basic
requirements for an additive probability measure (i.e.,
that the probabilities assigned to a set of mutually exclusive
and exhaustive outcomes are individually between 0 and 1,
and sum to 1).
An important feature of QP theory is the distinction
between superposition and basis states. In the abovementioned example, after the person has decided that she is
happy, then the state vector is |Ψ〉 = |happy〉; alternatively
if she decides that she is unhappy, then |Ψ〉 = |unhappy〉.
These are called basis states, with respect to the question
about happiness, because the answer is certain when the
state vector |Ψ〉 exactly coincides with one basis vector.
Note that this explains why the subspaces corresponding
to mutually exclusive outcomes (such as being happy and
being unhappy) are at right angles to each other. If a
person is definitely happy, i.e., |Ψ〉 = |happy〉, then we
want a zero probability that the person is unhappy, which
means a zero projection to the subspace for unhappy.
This will only be the case if the happy, unhappy subspaces
are orthogonal.
Before the decision, the state vector is a superposition of
the two possibilities of happiness or unhappiness, so that
|Ψ〉 = a|happy〉 + b|unhappy〉. The concept of superposition
differs from the CP concept of a mixed state. According
to the latter, the person is either exactly happy or exactly
unhappy, but we don’t know which, and so we assign
some probability to each possibility. However, in QP
theory, when a state vector is expressed as |Ψ〉 = a
|happy〉 + b|unhappy〉 the person is neither happy nor
unhappy. She is in an indefinite state regarding happiness,
simultaneously entertaining both possibilities, but being
uncommitted to either. In a superposition state, all we
can talk about is the potential or tendency that the
person will decide that she is happy or unhappy. Therefore,
a decision, which causes a person to resolve the indefinite
state regarding a question into a definite (basis) state, is
not a simple read-out from a pre-existing definite state;
instead, it is constructed from the current context and
question (Aerts & Aerts 1995). Note that other researchers
have suggested that the way of exploring the available premises can affect the eventual judgment, as much as the premises themselves, so that judgment is a constructive
process (e.g., Johnson et al. 2007; Shafer & Tversky
1985). The interesting aspect of QP theory is that it fundamentally requires a constructive role for the process of disambiguating a superposition state (this relates to the
Kochen–Specker theorem).
2.2. Compatibility
Suppose that we are interested in two questions, whether
the person is happy or not, and also whether the person
is employed or not. In this example, there are two outcomes with respect to the question about happiness, and
two outcomes regarding employment. In CP theory, it is
always possible to specify a single joint probability distribution over all four possible conjunctions of outcomes for
happiness and employment, in a particular situation. (Griffiths [2003] calls this the unicity principle, and it is fundamental in CP theory). By contrast, in QP theory, there is
a key distinction between compatible and incompatible
questions. For compatible questions, one can specify a
joint probability function for all outcome combinations
and in such cases the predictions of CP and QP theories
converge (ignoring dynamics). For incompatible questions,
it is impossible to determine the outcomes of all questions
concurrently. Being certain about the outcome of one
question induces an indefinite state regarding the outcomes
of other, incompatible questions.
This absolutely crucial property of incompatibility is one
of the characteristics of QP theory that differentiates it
from CP theory. Psychologically, incompatibility between
questions means that a cognitive agent cannot formulate
a single thought for combinations of the corresponding outcomes. This is perhaps because that agent is not used to
thinking about these outcomes together, for example, as in
the case of asking whether Linda (Tversky & Kahneman
1983) can be both a bank teller and a feminist. Incompatible
questions need to be assessed one after the other. A heuristic
guide of whether some questions should be considered
compatible is whether clarifying one is expected to interfere
with the evaluation of the other. Psychologically, the
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intuition is that considering one question alters our state of
mind (the context), which in turn affects consideration of
the second question. Therefore, probability assessment in
QP theory can be (when we have incompatible questions)
order and context dependent, which contrasts sharply with
CP theory.
Whether some questions are considered compatible or
incompatible is part of the analysis that specifies the corresponding cognitive model. Regarding the questions for
happiness and employment for the hypothetical person,
the modeler would need to commit a priori as to whether
these are compatible or incompatible. We consider in
turn the implications of each approach.
2.2.1. Incompatible questions. For outcomes corresponding to one-dimensional subspaces, incompatibility means
that subspaces exist at nonorthogonal angles to each
other, as in, for example, for the happy and employed subspaces in Figure 1b. Because of the simple relation we
assume to exist between happiness and employment, all
subspaces can be coplanar, so that the overall vector
space is only two dimensional. Also, recall that certainty
about a possible outcome in QP theory means that the
state vector is contained within the subspace for the
outcome. For example, if we are certain that the person
is happy, then the state vector is aligned with the happy
subspace. However, if this is the case, we can immediately
see that we have to be somewhat uncertain about the
person’s employment (perhaps thinking about being
happy makes the person a bit anxious about her job). Conversely, certainty about employment aligns the state vector
with the subspace for employed, which makes the person
somewhat uncertain about her happiness (perhaps her
job is sometimes stressful). This is a manifestation of the
famous Heisenberg uncertainty principle: Being clear on
one question forces one to be unclear on another incompatible question.
Because it is impossible to evaluate incompatible questions concurrently, quantum conjunction has to be
defined in a sequential way, and so order effects may
arise in the overall judgment. For example, suppose that
the person is asked first whether she is employed, and
then whether she is happy, that is, we have
Prob(happy|employed) = Phappy |cemployed l2
which leads to
Prob(employed ^ then happy) = Phappy Pemployed |cl2
Therefore, in QP theory, a conjunction of incompatible
questions involves projecting first to a subspace corresponding to an outcome for the first question and,
second, to a subspace for the second question (Busemeyer
et al. 2011). This discussion also illustrates the QP definition for conditional probability, which is in general
Prob(A|B) =
PA PB |cl2 Prob(B ^ then A)
=
Prob(B)
PB |cl2
(this is called Lüder’s law).
It is clear that the definition of conditional probability in
QP theory is analogous to that in CP theory, but for potential order effects in the sequential projection PAPB, when A
and B are incompatible.
The magnitude of a projection depends upon the angle
between the corresponding subspaces. For example,
when the angle is large, a lot of amplitude is lost between
successive projections. As can be seen in Figure 1b,
Phappy |cl2 , Phappy Pemployed |cl2
that is, the direct projection to the happy subspace (green
line) is less than the projection to the happy subspace via
the employed one (light blue line). (Color versions of the
figures in this article are available at http://dx.doi.org/10.
1017/S0140525X12001525].) The psychological intuition
would be that if the person is asked whether she is
employed or not, and concludes that she is, perhaps this
makes her feel particularly good about herself, which
makes it more likely that she will say she is happy. In classical terms, here we have a situation whereby
Prob(happy) , Prob(happy ^ employed)
Prob(employed ^ then happy) = Prob(employed)
· Prob(happy|employed)
whereby the first term is
Prob(employed) = Pemployed |cl2
The second term is the probability that the person is
happy, given that the person is employed. Certainty that
the person is employed means that the state vector is
cemployed l =
pemployed |cl
pemployed |cl
Therefore
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which is impossible in CP theory. Moreover, consider the
comparison between first asking “are you employed” and
then “are you happy” versus first asking “are you happy”
and then “are you employed.” In CP theory, this corresponds to
Prob(employed ^ happy) = Prob(happy ^ employed).
However, in QP theory conjunction of incompatible
questions fails commutativity. We have seen that
Prob(employed ^ then happy) = Phappy Pemployed |cl2
is large. By contrast,
Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
Prob(happy ^ then employed) = Pemployed Phappy |cl2
is less large, because in this case we project from |Ψ〉 to
|happy〉, whereby we lose quite a bit of amplitude (their
relative angle is large) and then from |happy〉 to |employed〉
(we lose more amplitude).
In general, the smaller the angle between the subspaces
for two incompatible outcomes, the greater the relation
between the outcomes. A small angle is analogous to a
high correlation in a classical framework. When there is a
small angle, a sequential projection of the state vector
from one subspace to the other loses little amplitude.
Accordingly, accepting one outcome makes the other
outcome very likely as well. The size of such angles and
the relative dimensionality of the subspaces are the cornerstones of QP cognitive models and are determined by the
known psychology of the problem. These angles (and the
initial state vector) have a role in QP theory analogous to
that of prior and conditional distributions in Bayesian modeling. In the toy illustration of Figure 1b, the only guidance
in placing the subspaces is that the employed and happy
subspaces should be near each other, to reflect the expectation that employment tends to relate to happiness. The
state vector was placed near the employed subspace,
assuming the person is confident in her employment.
Note that the above discussion does not concern probabilistic assessments indexed by time. That is, we are not
comparing
Prob(employed on Monday ^ happy on Tuesday)
versus
Prob(happy on Monday ^ employed on Tuesday).
Both CP and QP theories predict these to be different,
because the events are distinguished by time, so we no
longer compare the same events (“employed on Monday”
is not the same event as “employed on Tuesday”). Rather,
here we are concerned with the order of assessing a combination of two events, when the two events are defined in
exactly the same way. But could order dependence in
quantum theory arise as probability dependence in classical
theory? The answer is no because
Prob(A ^ B) = Prob(A)Prob(B|A) = Prob(B)Prob(A|B)
= Prob(B ^ A).
In quantum theory, the intermediate step is not possible
whenever PAPB = PBPA. Note that in an expressions such as
Prob(employed ^ then happy) = Phappy Pemployed |cl2
there are two sources of uncertainty. There is the classical
uncertainty about the various outcomes. There is a further
uncertainty as to how the state will collapse after the first
question (if the two questions are incompatible). This
second source of uncertainty does not exist in a classical framework, as classically it is assumed that a measurement (or
evaluation) simply reads off existing values. By contrast, in
quantum theory a measurement can create a definite value
for a system, which did not previously exist (if the state of
the system was a superposition one).
We have seen how it is possible in QP theory to have
definite knowledge of one outcome affect the likelihood
of an alternative, incompatible outcome. Order and
context dependence of probability assessments (and, relatedly, the failure of commutativity in conjunction) are some
of the most distinctive and powerful features of QP theory.
Moreover, the definitions for conjunction and conditional
probability in QP theory are entirely analogous to those
in CP theory, except for the potential of order effects for
incompatible questions.
2.2.2. Compatible questions. Now assume that the happiness and employment questions are compatible, which
means that considering one does not influence consideration of the other, and all four possible conjunctions of
outcomes are defined. To accommodate these outcome
combinations, we need a four-dimensional space, in
which each basis vector corresponds to a particular combination of happiness and employment outcomes
(Figure 1c is a three-dimensional simplification of this
space, leaving out the fourth dimension). Then, the probability that the person is happy and employed is given by
projecting the state vector onto the corresponding basis
vector. Clearly,
Prob(happy ^ employed) = Phappy ^ employed |cl2
= Prob(employed ^ happy).
Thus, for compatible questions, conjunction is commutative, as in CP theory.
The vector space for compatible outcomes is formed by
an operation called a tensor product, which provides a way
to construct a composite space out of simpler spaces. For
example, regarding happiness we can write
|Hl = h · |happyl + h′ · | happyl
and this state vector allows us to compute the probability
that the person is happy or not. Likewise, regarding
employment, we can write
|El = e · |employedl + e′ · | employedl.
As long as happiness and employment are compatible,
the tensor product between |H〉 and |E〉 is given by
|product statel = |Hl ⊗ |El
= h · e · |happyl ⊗ |employedl + h · e′ · |happyl
⊗ | employedl + h′ · e · | happyl ⊗ |employedl
+ h′ · e′ · | happyl ⊗ | employedl.
This four-dimensional product state is formed from the
basis vectors representing all possible combinations of
whether the person is employed or not and is happy
or not. For example, |happyl ⊗ |employedl| or for brevity
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|happy〉|employed〉, denotes a single basis vector that represents the occurrence of the conjunction “happy and
employed” (Figure 1c). The joint probability that the
person is employed and happy simply equals |h·e|2. This
probability agrees with the classical result for Prob
(employed ∧ happy), in the sense that the QP conjunction
is interpreted (and has the same properties) as conjunction
in CP theory.
What are the implications for psychological modeling?
Tensor product representations provide a concrete and rigorous way of creating structured spatial representations in
QP theory. Several researchers have pointed out that representations for even the most basic concepts must be
structured, as information about the different elements of
a concept are compared to like (alignable) elements in an
alternative concept (Goldstone 1994; Hahn et al. 2003;
Markman & Gentner 1993). Such intuitions can be
readily realized in a QP framework through tensor
product representations. Note that this idea is not new:
others have sought to develop structured representations
via tensor products (Smolensky 1990). The advantage of
QP theory is that a tensor product representation is supported by a framework for assessing probabilities.
CP theory is also consistent with structured representations. However, in QP theory, because of the property
of superposition, creating structured representations sometimes leads to a situation of entanglement. Entanglement
relates to some of the most puzzling properties of QP
theory. To explain it, we start from a state that is not
entangled, the |product state〉 described earlier, and
assume that the person is definitely employed (e=1), so
that the state reduces to
|reduced statel = h · |happyl|employedl
+ h′ · |happyl|employedl.
So far, we can see how the part for being happy is completely separate from the part for being employed. It
should be clear that in such a simple case, the probability
of being happy is independent (can be decomposed from)
the probability of being employed. As long as the state
vector has a product form (e.g., as mentioned), the components for each subsystem can be separated out. This situation is entirely analogous to that in CP theory for
independent events, whereby a composite system can
always be decomposed into the product of its separate
subsystems.
An entangled state is one for which it is not possible to
write the state vector as a tensor product between two
vectors. Suppose we have
|entangled statel = x · |happyl|employedl
+ w · |happyl|employedl.
This |entangled state〉 does not correspond to either a
decision being made regarding being happy or a clarification regarding employment. Such states are called
entangled states, because an operation that influences
one part of the system (e.g., being happy), inexorably
affects the other (clarifying employment). In other words,
in such an entangled state, the possibilities of being
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happy and employed are strongly dependent upon each
other. The significance of entanglement is that it can lead
to an extreme form of dependency between the outcomes
for a pair of questions, which goes beyond what is possible
in CP theory. In classical theory, one can always construct a
joint probability Prob(A,B,C) out of pairwise ones, and
Prob(A,B), Prob(A,C), and Prob(B,C) are all constrained
by this joint. However, in QP theory, for entangled
systems, it is not possible to construct a complete joint,
because the pairwise probabilities can be stronger than
what is allowed classically (Fine 1982).
2.3. Time evolution
So far, we have seen static QP models, whereby we assess the
probability for various outcomes for a state at a single point in
time. We next examine how the state can change in time.
Time evolution in QP theory involves a rotation (technically,
a unitary) operator (the solution to Schrödinger’s equation).
This dynamic operator evolves the initial state vector,
without changing its magnitude. It is important to recall
that the state vector is a superposition of components along
different basis vectors. Therefore, what evolves are the amplitudes along the different basis vectors. For example, a
rotation operator might move the state |Ψ〉 away from the |
happy〉 basis vector toward the |unhappy〉 one, if the
modeled psychological process causes unhappiness with
time. Analogously, time evolution in CP theory involves a
transition matrix (the solution to Kolmogorov’s forward
equation). The classical initial state corresponds to a joint
probability distribution over all combinations of outcomes.
Time evolution involves a transformation of these probabilities, without violating the law of total probability.
In both CP and QP theories, time evolution corresponds
to a linear transformation of the initial state. In CP theory,
the time-evolved state directly gives the probabilities for
the possible outcomes. Time evolution is a linear transformation that preserves the law of total probability. By
contrast, in QP theory, whereas the state vector amplitudes
are linearly transformed, probabilities are obtained by
squaring the length of the state vector. This nonlinearity
means that the probabilities obtained from the initial
state vector may obey the law of total probability, but this
does not have to be the case for the time-evolved ones.
Therefore, in QP theory, time evolution can produce probabilities that violate the law of total probability. This is a
critical difference between CP and QP theory and argues
in favor of the latter, to the extent that there are cognitive
violations of the law of total probability.
As an example, suppose the hypothetical person is due a
major professional review and she is a bit anxious about
continued employment (so that she is unsure about
whether she is employed or not). Prior to the review, she
contemplates whether she is happy to be employed or
not. In this example, we assume that the employment
and happiness questions are compatible (Figure 1c). In
CP theory, the initial probabilities satisfy
Prob(happy, unknown empl.) = Prob(happy ^ employed)
+ Prob(happy ^ not employed).
Next, assume that the state vector evolves for time t. This
process of evolution could correspond, for example, to the
Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
thought process of considering happiness, depending upon
employment assumptions. It would lead to a final set of
probabilities that satisfy
Prob(happy, unknown empl., at t)
= Prob(happy at t ^ employed)
+ Prob(happy at t ^ notemployed)
Although the final distribution differs from the initial distribution, they both obey the law of total probability. In QP
theory, we can write the initial state vector as
State(happy, unknown empl.) = State(happy ^ employed)
+ (happy ^ not employed).
After time evolution, we have
State(happy, unknownempl., at t)
= State(happy at t ^ employed)
+ State(happy at t ^ not employed)
but
Prob(happy, unknown empl., at t)
= Prob(happy at t ^ employed)
+ Prob(happy at t ^ not employed)
+ Interference(crossproduct) terms
(see Appendix). One way in which interference effects
can arise in QP theory is by starting with a state vector
that is a superposition of orthogonal states. Then, time evolution can result in the state vector being a superposition of
states, which are no longer orthogonal. As quantum probabilities are determined from the state vector by squaring
its length, we have a situation analogous to |a + b|2 = a 2 +
b 2 + a∗ b + b∗ a. When the states corresponding to a, b are
orthogonal, the interference terms a∗ b + b∗ a disappear
and QP theory reduces to CP theory. Otherwise, QP
theory can produce violations of the law of total probability.
Interference terms can be positive or negative and their
particular form will depend upon the specifics of the corresponding model. In the previous example, negative interference terms could mean that the person may think she would
be happy if it turns out she is employed (perhaps because of
the extra money) or that she would be happy if she loses her
job (perhaps she doesn’t like the work). However, when she
is unsure about her employment, she becomes unhappy. It
is as if these two individually good reasons for being happy
cancel each other out (Busemeyer & Bruza 2012, Ch. 9).
That a preference that is dominant under any single definite
condition can be reversed in an unknown condition is a
remarkable feature of QP theory and one that (as will be discussed) corresponds well to intuition about psychological
process (Tversky & Shafir 1992).
Suppose that the hypothetical person knows she will find
out whether she will be employed or not, before having the
inner reflection about happiness (perhaps she plans to think
about her happiness after a professional review). The
resolution regarding employment eliminates any possible
interference effects from her judgment, and the quantum
prediction converges to the classical one (Appendix).
Therefore, in QP theory, there is a crucial difference
between (just) uncertainty and superposition and it is
only the latter that can lead to violations of the law of
total probability. In quantum theory, just the knowledge
that an uncertain situation has been resolved (without
necessarily knowing the outcome of the resolution) can
have a profound influence on predictions.
3. The empirical case for QP theory in psychology
In this section, we explore whether the main characteristics
of QP theory (order/context effects, interference, superposition, entanglement) provide us with any advantage in
understanding psychological processes. Many of these situations concern Kahneman and Tversky’s hugely influential
research program on heuristics and biases (Kahneman et al.
1982; Tversky & Kahneman 1973; 1974; 1983), one of the
few psychology research programs to have been associated
with a Nobel prize (in economics, for Kahneman in 2002).
This research program was built around compelling demonstrations that key aspects of CP theory are often violated
in decision making and judgment. Therefore, this is a
natural place to start looking for whether QP theory may
have an advantage over CP theory.
Our strategy is to first discuss how the empirical finding in
question is inconsistent with CP theory axioms. This is not to
say that some model broadly based on classical principles
cannot be formulated. Rather, that the basic empirical
finding is clearly inconsistent with classical principles and
that a classical formalism, when it exists, may be contrived.
We then present an illustration for how a QP approach can
offer the required empirical coverage. Such illustrations
will be simplifications of the corresponding quantum models.
3.1. Conjunction fallacy
In a famous demonstration, Tversky and Kahneman (1983)
presented participants with a story about a hypothetical
person, Linda, who sounded very much like a feminist. Participants were then asked to evaluate the probability of statements about Linda. The important comparison concerned
the statements “Linda is a bank teller” (extremely unlikely
given Linda’s description) and “Linda is a bank teller and a
feminist.” Most participants chose the second statement as
more likely than the first, thus effectively judging that
Prob(bank teller) , Prob(bank teller ^ feminist).
This critical empirical finding is obtained with different
kinds of stories or dependent measures (including betting
procedures that do not rely on the concept of probability;
Gavanski & Roskos-Ewoldsen 1991; Sides et al. 2002;
Stolarz-Fantino et al. 2003; Tentori & Crupi 2012; Wedell
& Moro 2008). However, according to CP theory this is
impossible, because the conjunction of two statements can
never be more probable than either statement individually
(this finding is referred to as the conjunction fallacy). The
CP intuition can be readily appreciated in frequentist
terms: in a sample space of all possible Lindas, of the
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ones who are bank tellers, only a subset will be both bank
tellers and feminists. Tversky and Kahneman’s explanation
was that (classical) probability theory is not appropriate
for understanding such judgments. Rather, such processes
are driven by a similarity mechanism, specifically a representativeness heuristic, according to which participants
prefer the statement “Linda is a bank teller and a feminist”
because Linda is more representative of a stereotypical feminist. A related explanation, based on the availability heuristic, is that the conjunctive statement activates memory
instances similar to Linda (Tversky & Koehler 1994).
QP theory provides an alternative way to understand the
conjunction fallacy. In Figure 2, we specify |Ψ〉, the initial
state vector, to be very near the basis vector for |feminist〉
and nearly orthogonal to the basis vector for |bank teller〉.
Also, the |feminist〉 basis vector is neither particularly close
nor particularly far away from the |bank teller〉 one,
because to be a bank teller is not perhaps the most likely profession for feminists, but it is not entirely unlikely either.
These are our priors for the problem, that is, that the
description of Linda makes it very likely that she is a feminist
and very unlikely that she is a bank teller. Note the limited
flexibility in the specification of these subspaces and the
state vector. For example, the state vector could not be
placed in between the bank teller and feminist subspaces,
as this would mean that it is has a high projection to both
the bank teller and the feminist outcomes (only the latter
is true). Likewise, it would make no sense to place the feminist subspace near the bank teller one, or to the not bank
teller one, as feminism is a property that is largely uninformative as to whether a person is a bank teller or not.
Consider the conjunctive statement “Linda is a bank
teller and a feminist.” As we have seen, in QP theory,
conjunctions are evaluated as sequences of projections.
An additional assumption is made that in situations such
as this, the more probable possible outcome is evaluated
first (this is a reasonable assumption, as it implies that
more probable outcomes are prioritized in the decision
making process; cf. Gigerenzer & Todd 1999). Therefore,
the conjunctive statement involves first projecting onto the
feminist basis vector, and subsequently projecting on
the bank teller one. It is immediately clear that this
sequence of projections leads to a larger overall amplitude
(green line), compared to the direct projection from |Ψ〉
onto the bank teller vector.
Psychologically, the QP model explains the conjunction
fallacy in terms of the context dependence of probability
assessment. Given the information participants receive
about Linda, it is extremely unlikely that she is a bank
teller. However, once participants think of Linda in more
general terms as a feminist, they are more able to appreciate that feminists can have all sorts of professions, including
being bank tellers. The projection acts as a kind of abstraction process, so that the projection onto the feminist subspace loses some of the details about Linda, which
previously made it impossible to think of her as a bank
teller. From the more abstract feminist point of view, it
becomes a bit more likely that Linda could be a bank
teller, so that whereas the probability of the conjunction
remains low, it is still more likely than the probability for
just the bank teller property. Of course, from a QP
theory perspective, the conjunctive fallacy is no longer a
fallacy, it arises naturally from basic QP axioms.
Busemeyer et al. (2011) presented a quantum model
based on this idea and examined in detail the requirements
for the model to predict an overestimation of conjunction.
In general, QP theory does not always predict an overestimation of conjunction. However, given the details of the
Linda problem, an overestimation of conjunction necessarily follows. Moreover, the same model was able to
account for several related empirical findings, such as the
disjunction fallacy, event dependencies, order effects, and
unpacking effects (e.g., Bar-Hillel & Neter 1993; Carlson
& Yates 1989; Gavanski & Roskos-Ewoldsen 1991;
Stolarz-Fantino, et al. 2003). Also, the QP model is compatible with the representativeness and availability heuristics.
The projection operations used to compute probabilities
measure the degree of overlap between two vectors (or
subspaces), and overlap is a measure of similarity (Sloman
1993). Thus, perceiving Linda as a feminist allows the cognitive system to establish similarities between the initial
representation (the initial information about Linda) and
the representation for bank tellers. If we consider representativeness to be a similarity process, as we can do with
the QP model, it is not surprising that it is subject to chaining and context effects. Moreover, regarding the availability
heuristic (Tversky & Koehler 1994), the perspective from
the QP model is that considering Linda to be a feminist
increases availability for other related information about
feminism, such as possible professions.
3.2. Failures of commutativity in decision making
Figure 2. An illustration of the QP explanation for the
conjunction fallacy.
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We next consider failures of commutativity in decision
making, whereby asking the same two questions in different orders can lead to changes in response (Feldman &
Lynch 1988; Schuman & Presser 1981; Tourangeau et al.
Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
2000). Consider the questions “Is Clinton honest?” and “Is
Gore honest?” and the same questions in a reverse order.
When the first two questions were asked in a Gallup poll,
the probabilities of answering yes for Clinton and Gore
were 50% and 68%, respectively. The corresponding probabilities for asking the questions in the reverse order were,
by contrast, 57% and 60% (Moore 2002). Such order
effects are puzzling according to CP theory, because, as
noted, the probability of saying yes to question A and
then yes to question B equals
Prob(A) · Prob(B|A) = Prob(A ^ B) = Prob(B ^ A)
= Prob(B) · Prob(A|B).
deciding that Clinton is honest first, reduces the probability
that Gore is judged as honest).
The actual QP theory model developed for such failures
in commutativity was based on the abovementioned idea,
but was more general, so as to provide a parameter free
test of the relevant empirical data (e.g., there are various
specific types of order effects; Wang & Busemeyer, in
press).
A related failure of commutativity concerns the order of
assessing different pieces of evidence for a particular
hypothesis. According to CP theory, the order in which evidence A and B is considered, in relation to a hypothesis H,
is irrelevant, as
Prob(H|A ^ B) = Prob(H|B ^ A).
However, there have been demonstrations that, in fact,
Therefore, CP theory predicts that the order of asking
two questions does not matter. By contrast, the explanation
for order effects in social psychology is that the first question activates thoughts, which subsequently affect consideration of the second question (Schwarz 2007).
QP theory can accommodate order effects in Gallup polls,
in a way analogous to how the conjunction fallacy is
explained. In both cases, the idea is that the context for assessing the first question influences the assessment of any subsequent questions. Figure 3 is analogous to Figure 2. In
Figure 3, there are two sets of basis vectors, one for evaluating whether Clinton is honest or not and another for evaluating whether Gore is honest or not. The two sets of basis
vectors are not entirely orthogonal; we assume that if a
person considers Clinton honest, then that person is a
little more likely to consider Gore to be honest as well,
and vice versa (as they ran for office together). The initial
state vector is fairly close to the |Gore yes〉 vector, but less
close to the |Clinton yes〉 basis vector, to reflect the information that Gore would be considered more honest than
Clinton. The length of the projection onto the |Clinton
yes〉 basis vector reflects the probability that Clinton is
honest. It can be seen that the direct projection is less, compared to the projection via the |Gore yes〉 vector. In other
words, deciding that Gore is honest increases the probability
that Clinton is judged to be honest as well (and, conversely,
Prob(H|A ^ B) = Prob(H|B ^ A)
(Hogarth & Einhorn 1992; Shanteau 1970; Walker et al.
1972). Trueblood and Busemeyer (2011) proposed a QP
model for two such situations, a jury decision-making task
(McKenzie et al. 2002) and a medical inference one
(Bergus et al. 1998). For example, in the medical task participants (all medical practitioners) had to make a decision
about a disease based on two types of clinical information.
The order of presenting this information influenced the
decision, with results suggesting that the information presented last was weighted more heavily (a recency effect).
Trueblood and Busemeyer’s (2011) model involved considering a tensor product space for the state vector, with one
space corresponding to the presence or absence of the
disease (this is the event we are ultimately interested in)
and the other space to positive or negative evidence, evaluated with respect to the two different sources of information (one source of information implies positive
evidence for the disease and the other negative evidence).
Considering each source of clinical information involved a
rotation of the state vector, in a way reflecting the impact
of the information on the disease hypothesis. The exact
degree of rotation was determined by free parameters.
Using the same number of parameters, the QP theory
model produced better fits to empirical results than the
anchoring and adjustment model of Hogarth and Einhorn
(1992) for the medical diagnosis problem and for the
related jury decision one.
3.3. Violations of the sure thing principle
Figure 3.
An illustration of order effects in Gallup polls.
The model Trueblood and Busemeyer (2011) developed is
an example of a dynamic QP model, whereby the inference
process requires evolution of the state vector. This same
kind of model has been employed by Pothos and Busemeyer (2009) and Busemeyer et al. (2009) to account for
violations of the sure thing principle. The sure thing principle is the expectation that human behavior ought to
conform to the law of total probability. For example, in a
famous demonstration, Shafir and Tversky (1992) reported
that participants violated the sure thing principle in a oneshot prisoner’s dilemma task. This is a task whereby participants receive different payoffs depending upon whether
they decide to cooperate or defect, relative to another
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(often hypothetical) opponent. Usually the player does not
know the opponents’ move, but in some conditions Shafir
and Tversky told participants what the opponent had
decided to do. When participants were told that the
opponent was going to cooperate, they decided to defect;
and when they were told that the opponent was defecting,
they decided to defect as well. The payoffs were specified
in such a way so that defection was the optimal strategy.
The expectation from the sure thing principle is that,
when no information was provided about the action of
the opponent, participants should also decide to defect (it
is a “sure thing” that defection is the best strategy,
because it is the best strategy in all particular cases of
opponent’s actions). However, surprisingly, in the “no
knowledge” case, many participants reversed their judgment and decided to cooperate (Busemeyer et al. 2006a;
Croson 1999; Li & Taplin 2002). Similar results have
been reported for the two-stage gambling task (Tversky
& Shafir 1992) and a novel categorization–decisionmaking paradigm (Busemeyer et al. 2009; Townsend
et al. 2000). Therefore, violations of the sure thing principle
in decision making, although relatively infrequent, are not
exactly rare either. Note that this research has established
violations of the sure thing principle using within-participants designs.
Shafir and Tversky (1992) suggested that participants
perhaps adjust their beliefs for the other player’s action,
depending upon what they are intending to do (this principle was called wishful thinking and follows from cognitive
dissonance theory and related hypotheses, e.g., Festinger
1957; Krueger et al. 2012). Therefore, if there is a slight
bias for cooperative behavior, in the unknown condition
participants might be deciding to cooperate because they
imagine that the opponent would cooperate as well.
Tversky and Shafir (1992) described such violations of the
sure thing principle as failures of consequential reasoning.
When participants are told that the opponent is going to
defect, they have a good reason to defect as well, and, likewise, when they are told that the opponent is going to
cooperate. However, in the unknown condition, it is as if
these (separate) good reasons for defecting under each
known condition cancel each other out (Busemeyer &
Bruza 2011, Ch. 9).
This situation is similar to the generic example for violations of the law of total probability that we considered in
Section 2. Pothos and Busemeyer (2009) developed a
quantum model for the two-stage gambling task and prisoner’s dilemma embodying these simple ideas. A state vector
was defined in a tensor product space of two spaces, one
corresponding to the participant’s intention to cooperate
or defect and one for the belief of whether the opponent
is cooperating or defecting. A unitary operator was then
specified to rotate the state vector depending on the
payoffs, increasing the amplitudes for those combinations
of action and belief maximizing payoff. The same unitary
operator also embodied the idea of wishful thinking, rotating the state vector so that the amplitudes for the
“cooperate–cooperate” and “defect–defect” combinations
for participant and opponent actions increased. Thus, the
state vector developed as a result of two influences. The
final probabilities for whether the participant is expected
to cooperate or defect were computed from the evolved
state vector, by squaring the magnitudes of the relevant
amplitudes.
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Specifically, the probability of defecting when
the opponent is known to defect is based on the projection
Pparticipant to D |Ψopponent known D〉, where Pparticipant to D is a
projection operator corresponding to the participant choosing to defect. Similarly, the probability of defecting when
the opponent is known to cooperate is based on the projection Pparticipant to D |Ψopponent known C〉. But, in the unknown
case, the relevant state vector is the superposition
√1 |c
√1
opponent known D l + 2 |copponent known C l. The probability
2
for the participant to defect is computed by first using the
operator Pparticipant to D on this superposition, which gives
us Pparticipant to D (|Ψopponent known D〉 +|Ψopponent known C〉),
and subsequently squaring the length of the resulting projection. Therefore, we have another case of | a + b|2 = a 2 +
b 2 + a∗ b + b∗ a, with non-zero interference terms. Thus, a
high probability to defect in the two known conditions
(high a 2 and high b 2) can be offset by negative interference
terms, which means a lower probability to defect in the
unknown condition. We can interpret these computations
in terms of Tversky and Shafir’s (1992) description of the
result as a failure of consequential reasoning. Moreover,
the QP model provides a formalization of the wishful thinking hypothesis, with the specification of a corresponding
unitary operator matrix. However, note that this quantum
model is more complex than the ones considered previously. It requires more detail to see how interference
arises, in a way that leads to the required result, and the
model involves two parameters (model predictions are
robust across a wide range of parameter space).
3.4. Asymmetry in similarity
We have considered how the QP explanation for the conjunction fallacy can be seen as a formalization of the representativeness heuristic (Tversky & Kahneman 1983). This
raises the possibility that the QP machinery could be
employed for modeling similarity judgments. In one of
the most influential demonstrations in the similarity literature, Tversky (1977) showed that similarity judgments
violate all metric axioms. For example, in some cases, the
similarity of A to B would not be the same as the similarity
of B to A. Tversky’s (1977) findings profoundly challenged
the predominant approach to similarity, whereby objects
are represented as points in a multidimensional space,
and similarity is modeled as a function of distance. Since
then, novel proposals for similarity have been primarily
assessed in terms of how well they can cover Tversky’s
(1977) key empirical results (Ashby & Perrin 1988; Krumhansl 1978).
Pothos and Busemeyer (2011) proposed that different
concepts in our experience correspond to subspaces of
different dimensionality, so that concepts for which there
is more extensive knowledge were naturally associated
with subspaces of greater dimensionality. Individual
dimensions can be broadly understood as concept properties. They suggested that the similarity of a concept A to
another concept B (denoted, Sim (A,B)) could be
modeled with the projection from the subspace for the
first concept to the subspace for the second one: Sim (A,
B) = ||PB · PA · Ψ|| 2= Prob(A ∧ then B). Because in QP
theory probability is computed from the overlap between
a vector and a subspace, it is naturally interpreted as similarity (Sloman 1993). The initial state vector corresponds to
whatever a person would be thinking just prior to the
Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
comparison. This is set so that it is neutral with respect to
the A and B subspaces (i.e., prior to the similarity comparison, a participant would not be thinking more about A than
about B, or vice versa).
Consider one of Tversky’s (1977) main findings, that the
similarity of Korea to China was judged greater than the
similarity of China to Korea (actually, North Korea and
communist China; similar asymmetries were reported for
other countries). Tversky’s proposal was that symmetry is
violated, because we have more extensive knowledge
about China than about Korea, and, therefore, China has
more distinctive features relative to Korea. He was able
to describe empirical results with a similarity model
based on a differential weighting of the common and distinctive features of Korea and China. However, the only
way to specify these weights,was with free parameters
and alternative values for the weights, could lead to
either no violation of symmetry or a violation in a way opposite to the empirically observed one.
By contrast, using QP theory, if one simply assumes that
the dimensionality of the China subspace is greater than
the dimensionality of the Korea one, then a violation of symmetry in the required direction readily emerges, without the
need for parameter manipulation. As shown in Figure 4, in
the Korea to China comparison (4a), the last projection is
to a higher dimensionality subspace than is the last projection in the China to Korea comparison (4b). Therefore,
in the Korea to China case (4a), more of the amplitude of
the original state vector is retained, which leads to a prediction for a higher similarity judgment. This intuition was
validated with computational simulations by Pothos and
Busemeyer (2011), whose results indicate that, as long as
one subspace has a greater dimensionality than another, on
average the transition from the lower dimensionality subspace to the higher dimensionality one would retain more
amplitude than the converse transition (it has not been
proved that this is always the case, but note that participant
results with such tasks are not uniform).
3.5. Other related empirical evidence
Tversky and Kahneman are perhaps the researchers who
most vocally pointed out a disconnect between CP
models and cognitive process and, accordingly, we have
emphasized QP theory models for some of their most influential findings (and related findings). A skeptical reader
may ask, is the applicability of QP theory to cognition
mostly restricted to decision making and judgment?
Empirical findings that indicate an inconsistency with CP
principles are widespread across most areas of cognition.
Such findings are perhaps not as well established as the
ones reviewed previously, but they do provide encouragement regarding the potential of QP theory in psychology.
We have just considered a QP theory model for asymmetries in similarity judgment. Relatedly, Hampton (1988b,
Hampton1988 see also Hampton 1988a) reported an overextension effect for category membership. Participants
rated the strength of category membership of a particular
instance to different categories. For example, the rated
membership of “cuckoo” to the pet and bird categories
were 0.575 and 1 respectively. However, the corresponding
rating for the conjunctive category pet bird was 0.842, a
finding analogous to the conjunction fallacy. This paradigm
also produces violations of disjunction. Aerts and Gabora
Figure 4. Figure 4a corresponds to the similarity of Korea to
China and 4b to the similarity of China to Korea. Projecting to a
higher dimensionality subspace last (as in 4a) retains more of
the original amplitude than projecting onto a lower
dimensionality subspace last (as in 4b).
(2005b) and Aerts (2009) provided a QP theory account
of such findings. Relatedly, Aerts and Sozzo (2011b) examined membership judgments for pairs of concept combinations, and they empirically found extreme forms of
dependencies between concept combination pairs, which
indicated that it would be impossible to specify a complete
joint distribution over all combinations. These results could
be predicted by a QP model using entangled states to represent concept pairs.
In memory research, Brainerd and Reyna (2008) discovered an episodic overdistribution effect. In a training part,
participants were asked to study a set of items T. In test, the
training items T were presented together with related new
ones, R (and some additional foil items). Two sets of instructions were employed. With the verbatim instructions (V), participants were asked to identify only items from the set T.
With the gist instructions (G), participants were required to
select only R items. In some cases, the instructions
(denoted as V or G) prompted participants to select test
items from the T or R sets. From a classical perspective, as
a test item comes from either the T set or the R one, but
not both, it has to be the case that Prob(V|T) + Prob(G|T)=
Prob(VorG|T) (these are the probabilities of endorsing a
test item from the set T, as a function of different instructions). However, Brainerd and Reyna’s (2008) empirical
results were inconsistent with the classical prediction.
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Busemeyer and Bruza (2012, Ch. 6) explored in detail a range
of models for this memory overdistribution effect (apart from
a CP theory model, also a signal detection model, Brainerd
et al.’s [1999] dual process model, and a QP theory model).
The best performing models were the quantum model and
the dual process one, but the ability of the latter to cover
empirical results, in this case, perhaps depended too much
on an arbitrary bias parameter. Another example from
memory research is Bruza et. al.’s (2009) application of
quantum entanglement (which implies a kind of holism
inconsistent with classical notions of causality) to explain
associative memory findings, which cannot be accommodated
within the popular theory of spreading activation.
Finally, in perception, Conte et al. (2009) employed a
paradigm involving the sequential presentation of two ambiguous figures (each figure could be perceived in two different
ways) or the presentation of only one of the figures. It is possible that seeing one figure first may result in some bias in perceiving the second figure. Nonetheless, from a classical
perspective, one still expects the law of total probability to
be obeyed, so that p(A + ∧ B–) + p(A + ∧ B+) = p(A+) (A
and B refer to the two figures and the+and – signs to the
two possible ways of perceiving them). It turned out that
empirical results were inconsistent with the law of total probability, but a QP model could provide satisfactory coverage.
In other perception work, Atmanspacher et al. (2004; Atmanspacher & Filk 2010) developed and empirically tested a
quantum model that could predict the dynamic changes
produced during bistable perception. Their model provided
a picture of the underlying cognitive process radically different from the classical one. Classically, it has to be assumed
that at any given time a bistable stimulus is perceived with
a particular interpretation. In Atmanspacher et al.’s (2004)
model, by contrast, time periods of perception definiteness
were intermixed with periods in which the perceptual
impact from the stimulus was described with a superposition
state, making it impossible to consider it as conforming to a
particular interpretation. Atmanspacher et al.’s (2004)
model thus predicted violations of causality in temporal
continuity.
4. General issues for the QP models
4.1 Can the psychological relevance of CP theory be
disproved?
It is always possible to augment a model with additional
parameters or mechanisms to accommodate problematic
results. For example, a classical model could describe the
conjunction fallacy in the Linda story by basing judgment
not on the difference between a conjunction and an individual probability, but rather on the difference between
appropriately set conditional probabilities (e.g., Prob
(Linda|bank teller) vs. Prob(Linda|bank teller ∧ feminist);
cf. Tenenbaum & Griffiths 2001). Also, a conjunctive statement can always be conditionalized on presentation order,
so that one can incorporate the assumption that the last
piece of evidence is weighted more heavily than the first
piece. Moreover, deviations from CP predictions in judgment could be explained by introducing assumptions of
how participants interpret the likelihood of statements in
a particular hypothesis, over and above what is directly
stated (e.g., Sher & McKenzie 2008). Such approaches,
however, are often unsatisfactory. Arbitrary interpretations
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of the relevant probabilistic mechanism are unlikely to generalize to related empirical situations (e.g., disjunction fallacies). Also, the introduction of post-hoc parameters will
lead to models that are descriptive and limited in insight.
Thus, employing a formal framework in arbitrarily flexible
ways to cover problematic findings is possible, but of arguable explanatory value, and it also inevitably leads to criticism (Jones & Love 2011). But are the findings we
considered particularly problematic for CP theory?
CP theory is a formal framework; that is, a set of interdependent axioms that can be productively employed to lead
to new relations. Therefore, when obtaining psychological
evidence for a formal framework, we do not just support
the particular principles under scrutiny. Rather, such evidence corroborates the psychological relevance of all possible relations that can be derived from the formal
framework. For example, one cannot claim that one postulate from a formal framework is psychologically relevant,
but another is not, and still maintain the integrity of the
theory.
The ingenuity of Tversky, Kahneman, and their collaborators (Kahneman et al. 1982; Shafir & Tversky 1992;
Tversky & Kahneman 1973) was exactly that they provided
empirical tests of principles that are at the heart of CP
theory, such as the law of total probability and the relation
between conjunction and individual probabilities. Therefore, it is extremely difficult to specify any reasonable CP
model consistent with their results, as such models simply
lack the necessary flexibility. There is a clear sense that if
one wishes to pursue a formal, probabilistic approach for
the Tversky, Kahneman type of findings, then CP theory
is not the right choice, even if it is not actually possible to
disprove the applicability of CP theory to such findings.
4.2. Heuristics vs. formal probabilistic modeling
The critique of CP theory by Tversky, Kahneman and collaborators can be interpreted in a more general way, as a
statement that the attempt to model cognition with any
axiomatic set of principles is misguided. These researchers
thus motivated their influential program involving heuristics and biases. Many of these proposals sought to relate
generic memory or similarity processes to performance in
decision making (e.g., the availability and representativeness heuristics; Tversky & Kahneman 1983). Other
researchers have developed heuristics as individual computational rules. For example, Gigerenzer and Todd’s (1999)
“take the best” heuristic offers a powerful explanation of
behavior in a particular class of problem-solving situations.
Heuristics, however well motivated, are typically isolated: confidence in one heuristic does not extend to
other heuristics. Therefore, cognitive explanations based
on heuristics are markedly different from ones based on a
formal axiomatic framework. Theoretical advantages of
heuristic models are that individual principles can be examined independently from each other and that no commitment has to be made regarding the overall alignment of
cognitive process with the principles of a formal framework. Some theorists would argue that we can only understand cognition through heuristics. However, it is also often
the case that heuristics can be re-expressed in a formal way
or reinterpreted within CP or QP theory. For example, the
heuristics from the Tversky and Kahneman research
program, which were developed specifically as an
Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
alternative to CP models, often invoke similarity or
memory processes, which can be related to order/context
effects in QP theory. Likewise, failures of consequential
reasoning in prisoner’s dilemma (Tversky & Shafir 1992)
can be formalized with quantum interference effects.
The contrast between heuristic and formal probabilistic
approaches to cognition is a crucial one for psychology.
The challenge for advocates of the former is to specify
heuristics that cannot be reconciled with formal probability
theory (CP or QP). The challenge for advocates of the latter
is to show that human cognition is overall aligned with the
principles of (classical or quantum) formal theory.
4.3. Is QP theory more complex than CP theory?
We have discussed the features of QP theory, which distinguish it from CP theory. These distinctive features typically emerge when considering incompatible questions. We
have also stated that QP theory can behave like CP theory
for compatible questions (sect. 2.2.2). Accordingly, there
might be a concern that QP theory is basically all of CP
theory (for compatible questions) and a bit more, too (for
incompatible ones), so that it provides a more successful
coverage of human behavior simply because it is more
flexible.
This view is incorrect. First, it is true that QP theory for
compatible questions behaves a lot like CP theory. For
example, for compatible questions, conjunction is commutative, Lüder’s law becomes effectively identical to Bayes’s
law, and no overestimation of conjunction can be predicted. However, CP and QP theories can diverge, even
for compatible questions. For example, quantum timedependent models involving compatible questions can
still lead to interference effects, which are not possible in
classical theory (sect. 2.3). Although CP and QP theories
share the key commonality of being formal frameworks
for probabilistic inference, they are founded on different
axioms and their structure (set theoretic vs. geometric) is
fundamentally different. QP theory is subject to several
restrictive constraints; however, these are different from
the ones in CP theory.
For example, CP Markov models must obey the law of
total probability, whereas dynamic QP models can violate
this law. However, dynamic QP models must obey the
law of double stochasticity, while CP Markov models can
violate this law. Double stochasticity is a property of transition matrices that describes the probabilistic changes
from an input to an output over time. Markov models
require each column of a transition matrix to sum to
unity (so that they are stochastic), but QP models require
both each row and each column to sum to unity (so they
are doubly stochastic). Double stochasticity sometimes
fails and this rules out QP models (Busemeyer et al.
2009; Khrennikov 2010).
Moreover, QP models have to obey the restrictive law of
reciprocity, for outcomes defined by one-dimensional subspaces. According to the law of reciprocity, the probability
of transiting from one vector to another is the same as the
probability of transiting from the second vector to the first,
so that the corresponding conditional probabilities have to
be the same. Wang and Busemeyer (in press) directly
tested this axiom, using data on question order, and
found that it was upheld with surprisingly high accuracy.
More generally, a fundamental constraint of QP theory
concerns Gleason’s theorem, namely that probabilities
have to be associated with subspaces via the equation
Prob(A|c) = PA |cl2 .
Finding that Gleason’s theorem is psychologically
implausible would rule out quantum models. A critic may
wonder how one could test such general aspects of
quantum theory. Recently, however, Atmanspacher and
Römer (2012) were able to derive a test for a very
general property of QP theory (related to Gleason’s
theorem). Specifically, they proposed that failures of commutativity between a conjunction and one of the constituent elements of the conjunction (i.e., A vs. A ∧ B) would
preclude a Hilbert space representation for the corresponding problem. These are extremely general predictions and show the principled nature of QP theory
approaches to cognitive modeling.
Even if at a broad level CP and QP theories are subject to
analogous constraints, a critic may argue that it is still possible that QP models are more flexible (perhaps because of
their form). Ultimately, the issue of relative flexibility is a
technical one and can only be examined against particular
models. So far, there has only been one such examination
and, surprisingly, it concluded in favor of QP theory. Busemeyer et al. (2012) compared a quantum model with a traditional decision model (based on prospect theory) for a
large data set, from an experiment by Barkan and Busemeyer (2003). The experiment involved choices between
gambles, using a procedure similar to that used by
Tversky and Shafir (1992) for testing the sure thing principle. The models were equated with respect to the
number of free parameters. However, the models could
still differ with respect to their complexity. Accordingly,
Busemeyer et al. (2012) adopted a Bayesian procedure
for model comparison, which evaluates models on the
basis of both their accuracy and complexity. As Bayesian
comparisons depend upon priors over model parameters,
different priors were examined, including uniform and
normal priors. For both priors, the Bayes’s factor favored
the QP model over the traditional model (on average, by
a factor of 2.07 for normal priors, and by a factor of 2.47
for uniform priors).
Overall, QP theory does generalize CP theory in certain
ways. For example, it allows both for situations that are consistent with commutativity in conjunction (compatible
questions) and situations that are not (incompatible questions). However, QP theory is also subject to constraints
that do not have an equivalent in CP theory, such as
double stochasticity and reciprocity, and there is currently
no evidence that specific QP models are more flexible
than CP ones. The empirical question then becomes:
which set of general constraints is more psychologically relevant. We have argued that QP theory is ideally suited for
modeling empirical results that depend upon order/context
or appear to involve some kind of extreme dependence that
rules out classical composition. QP theory was designed by
physicists to capture analogous phenomena in the physical
world. However, QP theory does not always succeed, and
there have been situations in which the assumptions of
CP models are more in line with empirical results (Busemeyer et al. 2006). Moreover, in some situations, the
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predictions from QP and CP models converge, and in such
cases it is perhaps easier to employ CP models.
5. The rational mind
Beginning with Aristotle and up until recently, scholars have
believed that humans are rational because they are capable
of reasoning on the basis of logic. First, logic is associated
with an abstract elegance and a strong sense of mathematical correctness. Second, logic was the only system for
formal reasoning; therefore, scholars could not conceive
of the possibility that reasoning could be guided by an
alternative system. Logic is exactly this – logical – so how
could there be an alternative system for rational reasoning?
But this view turned out to be problematic. Considerable
evidence accumulated that naïve observers do not
typically reason with classical logic (Wason 1960); therefore,
classical logic could not be maintained as a theory of
thinking.
Oaksford and Chater (2007; 2009) made a compelling
case against the psychological relevance of classical logic.
The main problem is that classical logic is deductive, so
that once a particular conclusion is reached from a set of
premises, this conclusion is certain and cannot be altered
by the addition of further premises. Of course, this is
rarely true for everyday reasoning. The key aspect of everyday reasoning is its nonmonotonicity, as it is always possible
to alter an existing conclusion with new evidence. Oaksford
and Chater (2007; 2009) advocated a perspective of Bayesian rationality, which was partly justified using Anderson’s
(1990) rational analysis approach. According to rational
analysis, psychologists should look for the behavior function
that is optimal, given the goals of the cognitive agent and its
environment. Oaksford and Chater’s Bayesian rationality
view has been a major contribution to the recent prominence of cognitive theories based on CP theory. For
example, CP theories are often partly justified as rational
theories of the corresponding cognitive problems, which
makes them easier to promote than alternatives. For
example, in categorization, the rational model of categorization (e.g., Sanborn et al. 2010) has been called, well,
“rational.” By contrast, the more successful Generalized
Context Model (Nosofsky 1984) has received less corresponding justification (Wills & Pothos 2012).
There has been considerable theoretical effort to justify
the rational status of CP theory. We can summarize the
relevant arguments under three headings: Dutch book,
long-term convergence, and optimality. The Dutch book
argument concerns the long-term consistency of accepting
bets. If probabilities are assigned to bets in a way that goes
against the principles of CP theory, then this guarantees a
net loss (or gain) across time. In other words, probabilistic
assignment inconsistent with CP theory leads to unfair bets
(de Finetti et al. 1993). Long-term convergence refers to
the fact that if the true hypothesis has any degree of
non-zero prior probability, then, in the long run, Bayesian
inference will allow its identification. Finally, optimality is
a key aspect of Anderson’s (1990) rational analysis and
concerns the accuracy of probabilistic inference. According
to advocates of CP theory, this is the optimal way to
harness the uncertainty in our environment and make accurate predictions regarding future events and relevant
hypotheses.
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These justifications are not without problems. Avoiding a
Dutch book requires expected value maximization, rather
than expected utility maximization, that is, the decision
maker is constrained to use objective values rather than personal utilities, when choosing between bets. However,
decision theorists generally reject the assumption of objective value maximization and instead allow for subjective
utility functions (Savage 1954). This is essential, for
example, in order to take into account the observed risk
aversion in human decisions (Kahneman & Tversky 1979).
When maximizing subjective expected utility, CP reasoning
can fall prey to Dutch book problems (Wakker 2010). Longterm convergence is also problematic, because if the true
hypothesis has a prior probability of zero, it can never be
identified. This is invariably the case in Bayesian models,
as it is not possible to assign a non-zero probability to all candidate hypotheses. Overall, a priori arguments, such as the
Dutch book or long-term convergence, are perhaps appealing under simple, idealized conditions. However, as soon as
one starts taking into account the complexity of human cognition, such arguments break down.
Perhaps the most significant a priori justification for the
rationality of CP theory concerns optimality of predictions.
If reasoning on the basis of CP theory is optimal, in the
sense of predictive accuracy, then this seems to settle the
case in favor of CP theory. For example, is it more accurate
to consider Linda as just a bank teller, rather than as a bank
teller and a feminist? By contrast, QP theory embodies a
format for probabilistic inference which is strongly perspective and context dependent. For example, Linda may not
seem like a bank teller initially, but from the perspective of
feminism such a property becomes more plausible.
However, optimality must be evaluated under the constraints
and limited resources of the cognitive system (Simon 1955).
The main problem with classical optimality is that it
assumes a measurable, objective reality and an omniscient
observer. Our cognitive systems face the problem of
making predictions for a vast number of variables that
can take on a wide variety of values. For the cognitive
agent to take advantage of classical optimality, it would
have to construct an extremely large joint probability distribution to represent all these variables (this is the principle
of unicity). But for complex possibilities, it is unclear as to
where such information would come from. For example, in
Tversky and Kahneman’s (1983) experiment we are told
about Linda, a person we have never heard of before. Classical theory would assume that this story generates a sample
space for all possible characteristic combinations for Linda,
including unfamiliar ones such as feminist bank teller. This
just doesn’t seem plausible, let alone practical, considering
that for the bulk of available knowledge, we have no relevant experience. It is worth noting that Kolmogorov
understood this limitation of CP theory (Busemeyer &
Bruza 2012, Ch. 12). He pointed out that his axioms
apply to a sample space from a single experiment and
that different experiments require new sample spaces.
But his admonitions were not formalized, and CP modelers
do not take them into account.
Quantum theory assumes no measurable objective
reality; rather judgment depends on context and perspective. The same predicate (e.g., that Linda is a bank teller)
may appear plausible or not, depending upon the point of
view (e.g., depending on whether we accept Linda as a
feminist or not). Note that QP theory does assume
Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
systematic relations between different aspects of our
knowledge, in terms of the angle (and relative dimensionality) between different subspaces. However, each inference
changes the state vector, and, therefore, the perspective
from which all other outcomes can be evaluated. Note
also that context effects in QP theory are very different
from conditional probabilities in CP theory. The latter
are still assessed against a common sample space. With
the former, the sample space for a set of incompatible outcomes changes every time an incompatible question is evaluated (as this changes the basis for evaluating the state).
If we cannot assume an objective reality and an omniscient cognitive agent, then perhaps the perspective-driven
probabilistic evaluation in quantum theory is the best practical rational scheme. In other words, quantum inference is
optimal, for when it is impossible to assign probabilities to
all relevant possibilities and combinations concurrently.
This conclusion resonates with Simon’s (1955) influential
idea of bounded rationality, according to which cognitive
theory needs to incorporate assumptions about the computational burden which can be supported by the human
brain. For example, classically, the problem of assessing
whether Linda is a feminist and a bank teller requires the
construction of a bivariate joint probability space, which
assigns a probability density for each outcome regarding
these questions. By contrast, a QP representation is
simpler: it requires a univariate amplitude distribution for
each question, and the two distributions can be related
through a rotation. As additional questions are considered
(e.g., whether Linda might be tall or short) the efficiency
of the QP representation becomes more pronounced.
Note that classical schemes could be simplified by assuming
independence between particular outcomes. However,
independence assumptions are not appropriate for many
practical situations and will introduce errors in inference.
Note that the perspective dependence of probabilistic
assessment in QP theory may seem to go against an intuition that “objective” (classical) probabilities are somehow
more valid or correct. However, this same probabilistic
scheme does lead to more accurate predictions in the physical world, in the context of quantum physics. If the physical
world is not “objective” enough for CP theory to be used,
there is a strong expectation that the mental world, with
its qualities of flux and interdependence of thoughts,
would not be also.
The application of QP theory to cognition implies a
strong interdependence between thoughts, such that it is
typically not possible to have one thought without repercussions for other thoughts. These intuitions were extensively
elaborated in the work of Fodor (1983), with his proposals
that thought is isotropic and Quinean, so that revising or
introducing one piece of information can in principle
impact on most other information in our knowledge base.
Oaksford and Chater (2007; 2009) argued that it is
exactly such characteristics of thought that make CP
theory preferable to classical logic for cognitive modeling.
However, Fodor’s (1983) arguments also seem to go
against the neat reductionism in CP theory, required by
the principle of unicity and the law of total probability,
according to which individual thoughts can be isolated
from other, independent ones, and the degree of interdependence is moderated by the requirement to always
have a joint probability between all possibilities. QP
theory is not subject to these constraints.
Overall, accepting a view of rationality inconsistent with
classical logic was a major achievement accomplished by
CP researchers (e.g., Oaksford & Chater 2007; 2009).
For example, how can it be that in the Wason selection
task the “falsificationist” card choices are not the best
ones? Likewise, accepting a view of rationality at odds
with CP theory is the corresponding challenge for QP
researchers. For example, how could it not be that Prob
(A ∧ B)=Prob(B ∧ A)? The principles of CP theory have
been accepted for so long that they are considered self
evident. However, one of our objectives in Section 3 was
exactly to show how QP theory can lead to alternative,
powerful intuitions about inference, intuitions that emphasize the perspective-dependence of any probabilistic conclusion. We conclude with an interesting analogy.
Classical logic can be seen as a rational way of thinking,
but only in idealized situations in which deductive inference is possible, that is, such that there are no violations
of monotonicity. CP theory inference can also be seen as
rational, but only in idealized situations in which the
requirements from the principle of unicity match the capabilities of the observers (i.e., the possibilities that require
probabilistic characterization are sufficiently limited). For
the real, noisy, confusing, ever-changing, chaotic world,
QP is the only system that works in physics and, we strongly
suspect, in psychology as well.
6. Concluding comments
6.1 Theoretical challenges
The results of Tversky, Kahneman, and colleagues (e.g.,
Tversky & Kahneman 1974) preclude a complete explanation of cognitive processes with CP theory. We have
suggested that QP theory is the appropriate framework to
employ for cases in which CP theory fails. QP and CP theories are closely related and, also, the kind of models produced from CP and QP theories are analogous.
Therefore, it could be proposed that using CP and QP theories together, a complete explanation of cognitive processes would emerge.
In exploring such a proposal, the first step should be to
identify the precise boundary conditions between the applicability and failure of CP principles in cognitive modeling.
In other words, there is no doubt that in some cases cognitive process does rely on CP principles (perhaps the same
can also be said for classical logic principles). The results
of Tversky et al. (1974) reveal situations in which this
reliance breaks down. A key theoretical challenge concerns
understanding the commonalities between the experimental situations that lead to failures in the applicability of classical principles. For example, what exactly triggers the
relevance of a quantum probabilistic reasoning approach
in situations as diverse as the Linda problem, violations of
the sure thing principle in prisoner’s dilemma, and sets of
questions concerning United States presidential candidates? A preliminary suggestion is that perhaps the more
idealized the reasoning situation (e.g., is it cognitively feasible to apply the unicity principle?), the greater the psychological relevance of CP theory.
Another challenge concerns further understanding the
rational properties of quantum inference. The discussion
in Section 5 focused on the issue of accuracy, assuming
that the requirements from the principle of unicity have
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Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
to be relaxed. However, there is a further, potentially relevant literature on quantum information theory (Nielsen
& Chuang 2000), which concerns the processing advantages of probabilistic inference based on QP theory. For
example, a famous result by Grover (1997) shows how a
quantum search algorithm will outperform any classical
algorithm. The potential psychological relevance of such
results (e.g., in categorization theory) is an issue for much
further work (e.g., is it possible to approximate quantum
information algorithms in the brain?). These are exciting
possibilities regarding both the rational basis of quantum
cognitive models and the general applicability of quantum
theory to cognitive theory.
6.2. Empirical challenges
So far, the quantum program has involved employing
quantum computational principles to explain certain, prominent empirical findings. Such quantum models do not simply
– redescribe of results that have already had (some) compelling explanation. Rather, we discussed results that have presented ongoing challenges and have resisted explanation
based on classical principles. One objective for future work
is to continue identifying empirical situations that are problematic from a classical perspective.
Another objective is to look for new, surprising predictions, which take advantage of the unique properties of
quantum theory, such as superposition, incompatibility,
and entanglement. For example, Trueblood and Busemeyer (2011) developed a model to accommodate order
effects in the assessment of evidence in McKenzie et al.’s
(2002) task. The model successfully described data from
both the original conditions and a series of relevant extensions. Moreover, Wang and Busemeyer (in press) identified
several types of order effects that can occur in questionnaires, such as consistency and contrast (Moore 2002).
Their quantum model was able to make quantitative, parameter-free predictions for these order effects. In perception, Atmanspacher and Filk (2010) proposed an
experimental paradigm for bistable perception, so as to
test the predictions from their quantum model regarding
violations of the temporal Bell inequality (such violations
are tests of the existence of superposition states).
Overall, understanding the quantum formalism to the
extent that surprising, novel predictions for cognition can
be generated is no simple task (in physics, this was a
process that took several decades). The current encouraging results are a source of optimism.
6.3. Implications for brain neurophysiology
An unresolved issue is how QP computations are
implemented in the brain. We have avoided a detailed discussion of this research area because, although exciting, is
still in its infancy. One perspective is that the brain does
not instantiate any quantum computation at all. Rather,
interference effects in the brain can occur if neuronal membrane potentials have wave-like properties, a view that has
been supported in terms of the characteristics of electroencephalographic (EEG) signals (de Barros & Suppes 2009).
Relatedly, Ricciardi and Umezawa (1967), Jibu & Yasue
(1995), and Vitiello (1995) developed a quantum field
theory model of human memory, which still allows a classical description of brain activity. The most controversial
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(Atmanspacher 2004; Litt et al. 2006) perspective is that
the brain directly supports quantum computations. For
quantum computation to occur, a system must be isolated
from the environment, as environmental interactions
cause quantum superposition states to rapidly decohere
into classical states. Penrose (1989) and Hammeroff
(1998) suggested that microtubules prevent decoherence
for periods of time long enough to enable meaningful
quantum computation; in this view, the collapse of superposition states is associated with experiences of consciousness.
Overall, in cognitive science it has been standard to
initially focus on identifying the mathematical principles
underlying cognition, and later address the issue of how
the brain can support the postulated computations.
However, researchers have been increasingly seeking
bridges between computational and neuroscience models.
Regarding the QP cognitive program, this is clearly an
important direction for future research.
6.4. The future of QP theory in psychology
There is little doubt that extensive further work is essential
before all aspects of QP theory can acquire psychological
meaning. But this does not imply that current QP models
are not satisfactory. In fact, we argue that the quantum
approach to cognition embodies all the characteristics of
good cognitive theory: it is based on a coherent set of
formal principles, the formal principles are related to
specific assumptions about psychological process (e.g., the
existence of order/context effects in judgment), and it
leads to quantitative computational models that can parsimoniously account for both old and new empirical data.
The form of quantum cognitive theories is very much like
that of CP ones, and the latter have been hugely influential
in recent cognitive science. The purpose of this article is to
argue that researchers attracted to probabilistic cognitive
models need not be restricted to classical theory. Rather,
quantum theory provides many theoretical and practical
advantages, and its applicability to psychological explanation should be further considered.
APPENDIX
An elaboration of some of the basic definitions in QP theory
(See Busemeyer & Bruza 2012 for an extensive introduction)
Projectors (or projection operators)
Projectors are idempotent linear operators. For a onedimensional subspace, corresponding, for example, to the
|happy〉 ray, the projector is a simple outer product,
Phappy = |happy〉 〈happy|. Note that |happy〉 corresponds
to a column vector and 〈happy| denotes the corresponding
row vector (with conjugation).
Given the above subspace for “happy,” the probability
that a person is happy is given by
Phappy |cl2 = happyl khappy|cl2 .
In this expression, 〈happy|Ψ〉 is the standard dot (inner)
product and |happy〉 is a unit length vector. Therefore,
Phappy |cl2 = |khappy|cl|2 .
Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
The double lines on the left hand side of this equation
denote the length of the vector, so that ||Phappy |Ψ〉||2 is the
length of the vector squared. The single lines on the right
hand side denote the modulus (magnitude) of a complex
number. Therefore, in a real space, we can simply write
Thus, the effect of U(t) on a state vector is to rotate it in a
way that captures some dynamical aspect of the situation of
interest.
An example of how interference can arise in QP theory
2
Phappy |cl = (khappy|cl) .
2
Consider a situation whereby a person tries to assess
whether she is happy or not, depending upon whether
she is employed or not. We can write
Composite systems
Two subspaces can be combined into a composite space in
two ways: one way is by forming a tensor product space (as
in Figure 1b) and the other way is by forming a space from
a direct sum. First consider the formation of a tensor
product space. For example, suppose |happy〉, |∼happy〉
are two basis vectors that span the subspace H, representing the possibility of happiness, and suppose |employed〉,
|∼employed〉 are two basis vectors that span the subspace
E, representing the possibility of employment. Then, the
tensor product space equals the span of the four basis
vectors formed by the tensor products
{|happyl ⊗ |employedl, |happyl ⊗ |employedl,
|happyl ⊗ |employedl, |happyl ⊗ |employedl}.
Next consider the formation of a space by direct sum.
For example, suppose the subspace E is spanned by the
basis vectors
{|happyl ⊗ |employedl, |happyl ⊗ |employedl}
and suppose ∼E is the subspace spanned by the basis
vectors
1
1
|Initial statel = √ |cemployed l + √ |cemployed l
2
2
(this corresponds to a direct sum decomposition). Assume
that this initial state develops in time with a unitary matrix,
U(t), which could correspond to the thought process of
weighting the implications of being and not being employed
for happiness (Pothos & Busemeyer 2009; Trueblood &
Busemeyer 2011), so we end up with
1
1
|final statel = U(t) · √ |cemployed l + U(t) · √ |cemployed l.
2
2
Note that so far the situation is identical to what we would
have had if we were applying a CP theory Markov model.
The difference between the Markov CP model and the
QP one is in how probabilities are computed from the
time-evolved state. Consider the probability for being
happy; this can be extracted from a state by applying a projector operator M that picks out the coordinates for being
happy. In the QP case,
Prob(happy, unknown employment)
1
1
= M · U(t) · √ |cemployed l + M · U(t) · √ |cemployed l2
2
2
{|happyl ⊗ |employedl, |happyl ⊗ |employedl}.
Then the direct sum space is formed by all possible pairs
of vectors, one from E and another from ∼E.
Time dependence
The quantum state vector changes over time according to
Schrödinger’s equation,
d
|c(t)l = −i · H · |c(t)l
dt
where H is a Hermitian linear operator. This is the QP
theory equivalent of the Kolmogorov forward equation
for Markov models in CP theory. The solution to Schrödinger’s equation equals
c2 (t) = e−i·t·H · c1 = U(t) · c1
where H is a Hermitian operator and U(t) is a unitary one
(note that i 2=–1). The two (obviously related) operators H
and U(t) contain all the information about the dynamical
aspects of a system. The key property of U(t) is that it preserves lengths, so that
kU(t) · c|U(t) · Fl = kc|Fl.
That is, as has been discussed, probabilities are computed
from amplitudes through a squaring operation. This nonlinearity in QP theory can lead to interference terms that
produce violations of the law of total probability. Specifically,
Prob(happy, unknown employment)
= Prob(happy ^ employed) + Prob(happy^ employed)
+ Interference terms.
As the interference terms can be negative, the law of total
probability can be violated.
Suppose next that the person is determined to find out
whether she will be employed or not, before having this
inner reflection about happiness (perhaps she intends to
delay thinking about happiness until after her professional
review). Then, the state after learning about her employment will be either | Ψemployed〉 or | Ψ employed〉. Therefore,
what will evolve is one of these two states. Therefore, for
example,
Prob(happy|employed) = |M · U(t) · |cemployed l|2
and
Prob(happy|employed) = |M · U(t) · |c employed l|2 .
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Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
Overall, in this case,
Prob(happy, unknown employment)
= |M · U(t) · |cemployed l|2 · Prob(employed)
+ |M · U(t) · |c employed l|2 · Prob( employed).
It should be clear that in such a case there are no interference terms and the quantum result converges to the
classical one. Note that the “quantum” reasoner is still
uncertain about whether she will be employed. The
crucial difference is that in this case she knows she will
resolve the uncertainty regarding employment, before her
inner reflection. Therefore, regardless of the outcome
regarding employment, the evolved state will be a state
that is not a superposition one.
ACKNOWLEDGMENTS
We are grateful to Diederik Aerts, Harald Atmanspacher,
Thomas Filk, James Hampton, Mike Oaksford, Steven
Sloman, Jennifer Trueblood, and Christoph Weidemann
for their helpful comments. Jerome Busemeyer was supported by the grant NSF ECCS-1002188.
Open Peer Commentary
Quantum structure and human thought
doi:10.1017/S0140525X12002841
Diederik Aerts,a Jan Broekaert,a Liane Gabora,b and
Sandro Sozzoa
a
Center Leo Apostel, Departments of Mathematics and Psychology, Brussels
Free University, 1050 Brussels, Belgium; bDepartment of Psychology,
University of British Columbia, Okanagan Campus, Kelowna, BC V1V 1V7,
Canada.
[email protected]
http://www.vub.ac.be/CLEA/aerts
[email protected]
http://www.vub.ac.be/CLEA/Broekaert/
[email protected]
http://www.vub.ac.be/CLEA/liane
[email protected]
http://www.vub.ac.be/CLEA/people/sozzo/
Abstract: We support the authors’ claims, except that we point out that
also quantum structure different from quantum probability abundantly
plays a role in human cognition. We put forward several elements to
illustrate our point, mentioning entanglement, contextuality,
interference, and emergence as effects, and states, observables, complex
numbers, and Fock space as specific mathematical structures.
The authors convincingly demonstrate the greater potential of
quantum probability as compared with classical probability in
modeling situations of human cognition, giving various examples
to illustrate their analysis. In our commentary, we provide
additional arguments to support their claim and approach. We
want to point out, however, that it is not just quantum probability,
but also much more specific quantum structures, quantum states,
observables, complex numbers, and typical quantum spaces – for
example, Fock space – that on a deep level provide a modeling
of the structure of human thought itself.
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BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
A first insight about quantum structure in human cognition
came to us with the characterizations of classical and quantum
probability following from the hidden-variable investigation in
quantum theory – that is, the question of whether classical probability can model the experimental data of quantum theory (Bell
1964; Einstein et al. 1935). From these investigations, it follows
that when probability is applied generally to a physical system,
classical probability models the lack of knowledge of an underlying deterministic reality, whereas non-classical probability,
and possibly quantum probability, results when indeterminism
arises from the interaction between (measurement) context
and system, introducing genuine potentiality for the system
states (Aerts 1986). This allowed the identification of situations
in macroscopic reality entailing such non-classical indeterminism
and therefore being unable to be modeled by classical probability (Aerts 1986; Aerts et al. 1993). It shows that opinion
polls, where human decisions are intrinsically influenced by
the context, constitute such situations, and therefore entail
non-classical probability (Aerts & Aerts 1995). Our first argument to support and strengthen the authors’ claim is that a generalization of classical probability is necessary whenever
intrinsically contextual situations evoking indeterminism and
potentiality are present (Aerts 1986). We believe this to be commonly the case in situations of human cognition, and believe
quantum probability to be a plausible description for this indeterminism and potentiality.
Another result followed from studying the structure and
dynamics of human concepts themselves: how concepts
combine to form sentences and carry meaning in human
thought. An investigation into the relation of concepts to their
exemplars allowed for the devising of a Gedankenexperiment violating Bell’s inequalities, identifying the presence of quantum
entanglement (Aerts et al. 2000). Considering a combination of
concepts and its relation to exemplars led to an experimental violation of Bell’s inequalities, proving that concepts entangle when
they combine (Aerts & Sozzo 2011a). We studied the guppy
effect: Whereas a guppy is considered a very typical “pet-fish,” it
is regarded as neither a typical “pet” nor a typical “fish.” The
study of this effect, proved to be impossible to model with classical
fuzzy sets by Osherson and Smith (1981), led us to develop a
quantum-based concept theory presenting the guppy effect as a
non-classical context influence. Concepts are modeled as entities
in states in a complex Hilbert space, and quantities such as typicality are described by observables in the quantum formalism (Aerts
& Gabora 2005a; 2005b; Gabora & Aerts 2002). Our second argument to support and strengthen the authors’ approach is that next
to quantum effects such as entanglement and contextuality,
typical quantum representations of states and observables
appear in the combination dynamics of human concepts.
An abundance of experimental data violating set theoretical
and classical logic relations in the study of the conjunctions
and disjunctions of concepts (Hampton 1988b; 1998b) led to
the identification of new quantum effects – interference and
emergence – when these data were modeled using our
quantum concept formalism. Fock space, a special Hilbert
space also used in quantum field theory, turns out to constitute
a natural environment for these data. For the combination of
two concepts, the first sector of Fock space, mathematically
describing interference, corresponds to emergent aspects of
human thought, and the second sector, mathematically describing entanglement, corresponds to logical aspects of human
thought (Aerts 2007; 2009; 2011; Aerts & D’Hooghe 2009;
Aerts et al., in press). The quantum superposition in Fock
space, representing both emergent and logical thought,
models Hampton’s (1988a; 1988b) data well in our approach.
Our third argument to support and strengthen the authors’
analysis is that the quantum formalism, and many detailed
elements of its mathematical structure, for example, Fock
space, has proved to be relevant for the structure of human
thought itself.
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
Figure 1 (Aerts et al.). Part “F,” Part “V,” and Part “F or V” are a graphical representation of the relative membership weights of the
indicated exemplars with respect to “fruits,” “vegetables,” and “fruits or vegetables,” respectively. The light intensity at the spots where
the exemplars are located is a measure of the relative membership weight at that spot, and hence the graphs can be interpreted as light
sources passing through slits “F,” “V,” and “F and V.”
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
275
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
Table 1 (Aerts et al.). Relative membership weights of exemplars
with respect to fruits, vegetables and fruits or vegetables as
measured by Hampton (1988a)
At home in the quantum world
doi:10.1017/S0140525X12002853
Harald Atmanspacher
Vegetables
Fruits or
vegetables
0.0359
0.0425
0.0372
0.0586
0.0755
0.1026
0.1138
0.1184
0.0149
0.0138
0.0157
0.0167
0.0133
0.0108
0.0220
0.0269
0.0125
0.0170
0.0170
0.0688
0.0250
0.0255
0.0323
0.0446
0.0269
0.0249
0.0269
0.0415
0.0604
0.0555
0.0480
0.0670
0.0146
0.0165
0.0385
0.0323
0.0100
0.0140
0.0112
0.0095
0.0324
0.0301
0.0545
0.0658
0.0713
0.0788
0.0293
0.0604
0.0482
0.0338
0.0506
0.0533
0.0881
0.0797
0.0143
0.0140
0.0155
0.0127
0.0724
0.0679
0.0713
0.1284
0.0412
0.0266
0.0294
0.0541
0.0688
0.0579
0.0642
0.0248
0.0308
0.0222
Fruits
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Almond
Acorn
Peanut
Olive
Coconut
Raisin
Elderberry
Apple
Mustard
Wheat
Root ginger
Chili
pepper
Garlic
Mushroom
Watercress
Lentils
Green
pepper
Yam
Tomato
Pumpkin
Broccoli
Rice
Parsley
Black
pepper
We finish our commentary by presenting a graphic illustration
of the interference of concepts as it appears in our quantum
concept theory. Figure 1 represents the cognitive interference
of the two concepts “fruits” and “vegetables” combined in the disjunction “fruits or vegetables.” Part “F,” Part “V,” and Part “F or
V” illustrate the relative membership weights of the different
exemplars with respect to “fruits,” “vegetables,” and “fruits or vegetables,” respectively, measured in Hampton (1988a) and presented in Table 1. The illustration is built following standard
quantum theory in a Hilbert space of complex wave functions in
a plane. The exemplars are located at spots of the plane such
that the squares of the absolute values of the quantum wave functions for “fruits,” “vegetables” and “fruits or vegetables” coincide
with the relative membership weights measured.
The wave function for “fruits or vegetables” is the normalized
sum – that is, the superposition – of the two wave functions for
“fruits” and for “vegetables,” and hence the square of its absolute
value includes an interference term. The light intensity at the
spots where the exemplars are located is a measure of the relative
membership weight at that spot, which means that the graphs can
be seen as representations of light passing through slits, where
Part “F” corresponds to slit “F” open, Part “V” to slit “V” open,
and Part “F or V” to both slits “F” and “V” open. Hence, the
graphs illustrate the cognitive interference of “fruits or vegetables” in analogy with the famous double-slit interference
pattern of quantum theory (Feynman 1988). The interference
pattern is clearly visible (Part “F or V” of Fig. 1), and very
similar to well-known interference patterns of light passing
through an elastic material under stress. Mathematical details
can be found in Aerts et al. (in press).
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Institute for Frontier Areas of Psychology, D-79098 Freiburg, Germany;
Collegium Helveticum, CH-8092 Zurich, Switzerland.
[email protected]
http://www.igpp.de/english/tda/cv/cv_ha.htm
Abstract: One among many misleading quotations about the alleged
mysteries of quantum theory is from Feynman (1965): “I think I can
safely say that nobody understands quantum mechanics.” Today we
know that quantum theory describes many aspects of our world in a
fully intelligible fashion. Pothos & Busemeyer (P&B) propose ways in
which this may include psychology and cognitive science.
It was an old idea by Niels Bohr, one of the founding architects of
quantum physics, that central features of quantum theory, such as
complementarity, are also of pivotal significance beyond the
domain of physics. Bohr became familiar with the notion of complementarity through the psychologist Edgar Rubin and,
indirectly, William James (Holton 1970). Although Bohr always
insisted on the extraphysical relevance of complementarity, he
never elaborated this idea in concrete detail, and for a long time
no one else did so either. This situation has changed; there are
now a number of research programs applying key notions of
quantum theory beyond physics, in particular to psychology and
cognitive science.
The first steps in this direction were made by Aerts and collaborators in the early 1990s (Aerts & Aerts 1995) in the framework
of non-Boolean logic of incompatible (complementary) propositions.
Alternative ideas come from Khrennikov (1999), focusing on nonclassical probabilities, and Atmanspacher et al. (2002), proposing
an algebraic framework with non-commuting operations. More
recently, Bruza and colleagues as well as Busemeyer and colleagues
have moved this novel field of research even more into the center of
attention. The target article by Pothos & Busemeyer (P&B), and a
novel monograph by Busemeyer and Bruza (2012), reflect these
developments.
Intuitively, it is plausible that non-commuting operations or nonBoolean logic should be relevant, even inevitable, for mental
systems. The non-commutativity of operations simply means that
the sequence in which operations are applied matters for the
final result. This is so if individual mental states are assumed to
be dispersive (as individual quantum states are, as opposed to classical states). As a consequence, their observation amounts not only
to registering a value, but entails a backreaction changing the
observed state: something that seems evident for mental systems.
Non-Boolean logic refers to propositions that may have unsharp
truth values beyond “yes” or “no.” However, this is not the result
of subjective ignorance but must be understood as an intrinsic
feature. The proper framework for a logic of incompatible propositions is a partial Boolean lattice (Primas 2007), where locally
Boolean sublattices are pasted together in a globally non-Boolean
way – just like an algebra of generally non-commuting operations
may contain a subset of commuting operations.
Although these formal extensions are essential for quantum
theory, they have no dramatic effect on the way in which experiments are evaluated. The reason is that the measuring tools, even
in quantum physics, are typically Boolean filters, and, therefore, virtually all textbooks of quantum physics get along with standard probability theory à la Kolmogorov. Only if incompatible experimental
scenarios are to be comprehensively discussed in one single
picture do the pecularities provided by non-classical thinking
become evident and force us to leave outmoded classical reasoning.
In this sense, the authors use the notion of “quantum probability”
for psychological and cognitive models and their predictions (cf.
Gudder 1988; Redei & Summers 2007). As Busemeyer points out,
an experiment in psychology is defined as a collection of experimental conditions. Each one of them produces indivisible outcomes
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
forming a complete set of mutually exclusive events. Whereas Kolmogorov probabilities refer to events for a single condition, quantum
probabilities refer to the entire set of incompatible conditions,
necessary for a comprehensive description of the experiment.
In such a description, events are represented as subspaces of a
Hilbert space (as in quantum physics), and all subspaces correspond to orthogonal projectors. A state is defined as a vector,
and the probability of an event equals the squared length of the
projection of the state onto the subspace representing that
event. As all events under each single experimental condition
commute, they form a Boolean algebra and the probabilities
assigned to them satisfy the axiomatics of Kolmogorov.
However, all events of the entire experiment (i.e., the events of
all experimental conditions) only form a partial Boolean algebra
if some of them do not commute. And as Kolmogorov’s axioms
imply Bayes’ rule, Bayesian reasoning, very influential in psychology, will generally fail to describe experiments with incompatible
conditions properly.
Whereas the authors focus on decision theory, routes to be
explored in more detail include uncertainty relations, in which
order effects arise in variances in addition to mean shifts (Atmanspacher & Römer 2012). A key feature of quantum theory, entanglement as tested by Bell-type inequalities, has been suggested by
Atmanspacher and Filk (2010) for bistable perception and by
Bruza et al. (2012) for non-decomposable concept combinations.
Another possible move to incorporate complementarity and
entanglement in psychology is based on a state space description
of mental systems. If mental states are defined on the basis of cells
of a state space partition, then this partition needs to be well tailored to lead to robustly defined states. Ad hoc chosen partitions
will generally create incompatible descriptions (Atmanspacher &
beim Graben 2007) and states may become entangled (beim
Graben et al. 2013). This way it is possible to understand why
mental activity may exhibit features of quantum behavior
whereas the underlying neural dynamics are strictly classical.
A further important issue is the complexity or parsimony of
Hilbert space models as compared with classical (Bayesian,
Markov) models. Atmanspacher and Römer (2012) proposed an
option to test limitations of Hilbert space modeling by outcomes
of particular joint measurements. Such tests presuppose that the
situation under study is framed well enough to enable welldefined research questions; a requirement that must be carefully
observed to avoid superficial reasoning without sustainable
substance.
With the necessary caution, I am optimistic that this novel field
will grow from work in progress to an important subject area of
psychology. A quantum theoretically inspired understanding of
reality, including cognition, will force us to revise plugged-in
cliches of thinking and resist overly naive world views. The
Boolean “either-or” in logic and the law of commutativity in
elementary calculations are special cases with their own significance, but it would be wrong to think that their generalization
holds potential only for exotic particles and fields in physics.
The opposite is the case.
extension of the quantum probability formalism to psychophysical
measures, such as detectability and discriminability.
The Hilbert space formalism seems to be a suitable framework to
accommodate the experimental richness of cognitive phenomena.
The target article by Pothos & Busemeyer (P&B) accomplishes the
impressive task of providing, in as simple a way as possible, the
theoretical grounds as well as the empirical underpinnings of a
probabilistic model capable of grasping many aspects of human
cognition. The contribution of this commentary is to point out
that important concepts arising from signal detection theory
(SDT) can be easily recast into the language of quantum probability. If useful, this addition to P&B’s model might be used to
describe several phenomena involved in perceptual detectability
and discriminability, enlarging the theoretical reach of their proposal and offering new alternatives to verify its empirical content.
SDT is a powerful tool that has been very successful in many
areas of psychological research (Green & Swets 1966; Macmillan
& Creelman 2005). Originally stemming from applications of statistical decision theory to engineering problems, classical SDT has
been reframed over the years under many different assumptions
and interpretations (Balakrishnan 1998; DeCarlo 1998; Parasuraman & Masalonis 2000; Pastore et al. 2003; Treisman 2002).
However, irrespective of their formulation, signal detection theories rely on two fundamental performance measures: sensitivity
and bias. Whereas sensitivity refers to the ability of an observer to
detect a stimulus or discriminate between two comparable stimuli,
response bias implies a decision rule or criterion, which can favor
the observer’s response in one direction or another.
As in standard SDT and its sequels, in this commentary sensitivity and bias are also derived from the probabilities of hits, p
(H), and of false alarms, p(FA), defined as the probability of indicating the presence of a stimulus when it is present or absent. Two
other quantities, the probabilities of misses, p(M), and correct
rejections, p(CR), are complementary, respectively, to p(H) and
p(FA): p(H) + p(M) = 1 = p(FA) + p(CR).
To translate these concepts into the language of Hilbert space,
we need to represent a perceptual state by a state vector |Ψ〉. This
vector denotes the perceptual state of an observer immediately
after a trial wherein either the background noise was presented
in isolation (|Ψn〉) or the noisy background was superimposed
with the target stimulus (|Ψn+s〉). These state vectors are formed
by a linear combination of the |yes〉 and |no〉 vectors, an orthogonal vector basis in the present Hilbert space. The components
of a state vector along the one-dimensional subspaces that represent the two possible outcomes (either the response “yes” or
the response “no” to the question “was the stimulus present?”)
are given by the projection of the state vector onto the subspaces
for “yes” and “no,” which are spanned by the basis vectors (Fig. 1).
Denoting by Pyes and Pno the projection operators (projectors)
onto the subspaces spanned by the basis vectors, the components
C1 and C2 of a state vector |Ψ〉 = C1|yes〉 + C2|no〉 are given by
C1 = ||Pyes|Ψ〉|| and C2 = ||Pno|Ψ〉||.
As we can see in Figure 1, the probabilities of hits, misses, false
alarms, and correct rejections are obtained by the action of the
projectors Pyes and Pno on the state vectors |Ψn〉 and |Ψn+s〉.
These probabilities can be computed by means of the following
“statistical algorithm” (Hughes 1989):
Signal detection theory in Hilbert space
doi:10.1017/S0140525X12002865
Marcus Vinícius C. Baldo
Department of Physiology and Biophysics, “Roberto Vieira” Laboratory of
Sensory Physiology, Institute of Biomedical Sciences, University of São Paulo,
05508-900 São Paulo, SP, Brazil.
[email protected]
Abstract: The Hilbert space formalism is a powerful language to express
many cognitive phenomena. Here, relevant concepts from signal detection
theory are recast in that language, allowing an empirically testable
p(H) = kcn+s |Pyes cn+s l = Pyes |cn+s l2
p(FA) = kcn |Pyes cn l = Pyes |cn l
2
(eq. 1)
(eq. 2)
p(M) = kcn+s |Pno cn+s l = Pno |cn+s l
2
p(CR) = kcn |Pno cn l = Pno |cn l2
(eq. 3)
(eq. 4)
Analogously to SDT, two measures can be extracted from the
vector representation of a perceptual state: an angle, δ, which
evaluates the separation between the two state vectors, |Ψn〉 and
|Ψn+s〉, gives a measure of sensitivity; another angle, χ, which
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
277
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
Figure 1 (Baldo). A representation of perceptual states in
Hilbert space. The state vectors |Ψn〉 and |Ψn+s〉 represent the
perceptual state of an observer exposed, respectively, to noise
only or to a signal presented on a noisy background. On each
condition, the probability of the observer reporting the presence
(“yes”) or absence (“no”) of the signal is given by squaring the
projection of each vector onto the respective basis vector (see
equations 1–4).
evaluates the location of the state vectors as a whole in relation to
the basis vectors, gives a measure of response bias. In two dimensions, these quantities can be calculated from the pair of probabilities p(H) and p(FA):
d = arccos[p(FA)1/2 ] − arccos[p(H)1/2 ]
1/2
x = (1/2)(arccos[p(FA)
] + arccos[p(H)
some value, or it turns out to be only a mathematical exercise, is an
empirical question.
ACKNOWLEDGMENTS
This work was partially supported by the São Paulo Research Foundation
and the National Counsel for Scientific and Technological Development
(Brazil). I also thank Nestor Caticha for fruitful discussions on this topic.
(eq. 5)
1/2
]) − p/4
(eq. 6)
Similarly to SDT, the measures of sensitivity and bias in Hilbert
space are also, respectively, the subtraction and addition of terms
given by nonlinear transformations of hit and false alarm rates. In
Equation 6, the term –π/4 is added only to set χ = 0 for an
unbiased observer. Analogously to SDT, χ > 0 means a stricter criterion (a response bias toward lower hit and false alarm rates),
whereas χ < 0 means a more lax criterion (a bias toward higher
hit and false alarm rates). Figure 2, which resembles a receiver
operating characteristic (ROC) representation, shows a family of
isosensitivity curves in which the decision criterion χ changes as
a function of the hit and false alarm rates along five different magnitudes of the sensitivity measure, from δ = 0 to δ = π/3.
This geometric representation of SDT in Hilbert space can help
us visualize many perceptual phenomena. As one example, the
modulatory effect of attention could be conceived of as the
action of a unitary operator A on the noise-plus-signal state
vector, |Ψn+s〉, the noise-only state vector, |Ψn〉, or both, always
resulting in a rotation that, without changing their magnitudes,
moves them away from each other, thus increasing the sensitivity δ.
The physiological interpretation of the attentional operator A
would be the enhancement of the signal, the suppression of
noise, or both, depending on which vectors the operator’s action
takes place on. However, a unitary operator that acts on both
vectors at the same time, and whose action results in a rotation
that preserves their angular separation δ, can be viewed as a
change in the decision criterion χ only.
In conclusion, the idea of this commentary is to accommodate
in the language of Hilbert spaces some important concepts that
are extremely valuable in the experimental exploration of psychophysical and perceptual phenomena. Whether this attempt bears
278
Figure 2 (Baldo). A ROC curve in Hilbert space. Each line is an
iso-sensitivity curve that provides the proportion of hit and false
alarm rates along a constant sensitivity measure (δ). In the
present model, the sensitivity measure is given by the angle, δ,
that separates the two state vectors |Ψn〉 and |Ψn+s〉.
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
Can quantum probability help analyze the
behavior of functional brain networks?
doi:10.1017/S0140525X12002877
Arpan Banerjee and Barry Horwitz
Brain Imaging and Modeling Section, Voice, Speech and Language Branch,
National Institute on Deafness and other Communication Disorders, National
Institutes of Health, Bethesda, MD 20892-1402.
[email protected]
[email protected]
http://www.nidcd.nih.gov/research/scientists/pages/horwitzb.aspx
Abstract: Pothos & Busemeyer (P&B) argue how key concepts of quantum
probability, for example, order/context, interference, superposition, and
entanglement, can be used in cognitive modeling. Here, we suggest that
these concepts can be extended to analyze neurophysiological
measurements of cognitive tasks in humans, especially in functional
neuroimaging investigations of large-scale brain networks.
We agree in general with the views expressed in the target article
by Pothos & Busemeyer (P&B) about the application of quantum
probability (QP) to cognitive science. Moreover, we believe that
the mathematical framework of QP can be used as well to interpret brain network interactions underlying cognitive tasks, and
that this should be further explored in the future. Here, we
emphasize that the concept of functional brain network embodies
notions similar to those of QP, such as entanglement, interference, and incompatibility.
Modern neuroimaging experimental designs are often based on
the idea of studying a complex cognitive task as a sum of multiple
sensory/cognitive factors that have corresponding processing
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
modules in the brain (Sternberg 2011). For example, to study
auditory–visual integration, one would design unimodal control
tasks (factors) employing visual and auditory stimuli separately,
and examine the change of brain responses during presentation
of combined visual–auditory stimuli (Molholm et al. 2004).
Another example comes from investigating the neural correlates
of bimanual coordination: execution of movements with both
limbs, for example, two hands or two feet. Here, unimanual movements (movement with one limb at a time) serve as factors for
studying the neural correlates of bimanual movements (Banerjee
et al. 2012a; Debaere et al. 2001).
A fundamental problem with factor-based approaches is that
they implicitly assume that each cognitive component triggers
an additional neural correlate that is the same, irrespective of
the cognitive or physiological context (Banerjee et al. 2008). For
example, while studying multisensory integration, one often
assumes that the same brain response during an auditory stimulus
presentation is expected to be triggered in auditory areas when a
visual stimulus is tagged with the auditory stimulus presentation as
a combined auditory–visual stimulus (Molholm et al. 2004). The
reader of P&B will note that this assumption is identical to the
concept of “compatibility,” and is not appropriate for studying
the neural basis of human sensory and cognitive processes that
are adaptive to even subtle changes of context (McIntosh 2004).
Furthermore, an implicit assumption in factor-based neuoimaging designs is that any residual (e.g., bimanual: sum of left and
right unimanual) brain activity comes from recruitment of
additional brain areas related only to the missing factor not
addressed by the control tasks (Banerjee et al. 2008). Hence,
the possibility of “interference” of control-related networks
during the active cognitive task is ignored by factor-based
designs. This approach thus disregards the “degenerate” nature
of cognitive neural systems; different networks can coordinate
to perform the same tasks (Price & Friston 2002; Tononi et al.
1999). Degeneracy is an omnipresent phenomenon of complex
biological systems, and the search for measures to characterize
degenerate neural systems is an ongoing area of research
(Edelman & Gally 2001; Tononi et al. 1999). Current neuroscience research indicates that there are two distinct conceptualizations of neural degeneracy: structural and functional. The same
neural structures can give rise to different functions (structural)
and different neural structures can generate the same function
(functional) (McIntosh 2004; Price & Friston 2002). An example
of the former was observed in an electroencephalographic
(EEG) study of bimanual coordination (Banerjee et al. 2012a)
in which participants were asked to move their left or right
fingers independently and simultaneously in different trials at
specific frequencies by syncing with a rhythmic visual stimulus.
During execution of stable bimanual coordination patterns,
neural dynamics were dominated by temporal modulation of
unimanual networks. An example of functional degeneracy
occurred in an investigation of the temporal microstructure of a
long-term memory retrieval process (Banerjee et al. 2012b).
Here, magnetoencephalographic (MEG) data were used to
examine the network recruitment of auditory areas during a
visual–auditory paired associate task compared with a visual–
visual paired associate task. It was found that visual–visual
and visual–auditory memory recollection involved equivalent
network components without any additional recruitment during
an initial period of the sensory processing stage, which was then
followed by recruitment of additional network components for
modality-specific memory recollection.
P&B point out that the QP concept of “entanglement” plays a
crucial role in understanding cognitive behavior. The mathematical framework of expressing “entanglement” involves quantifying
the evolution of correlated state variables, and hence is pertinent
for the study of functional brain networks. Data acquired by functional neuroimaging (i.e., functional magnetic resonance imaging
[fMRI] and EEG/MEG) are being investigated by cognitive neuroscientists using a variety of network-based mathematical tools,
such as functional connectivity (Beckmann et al. 2005; Horwitz
et al. 1992), effective connectivity (Friston et al. 2003; McIntosh
et al. 1994), and graph theory (Bullmore & Sporns 2009). A critical
point with respect to functional networks is that as all nodes are
functionally linked to one another, any alteration in even a
single link results in alterations throughout the network (Kim &
Horwitz 2009). In essence, one can say that brain network
nodes (regions) that may not be anatomically connected and are
physically separated by large distances can become “entangled”
with one another to facilitate task execution. This aspect of
network behavior has important ramifications for the way in
which functional brain imaging data are interpreted in comparing
tasks, or patients and healthy subjects (Kim & Horwitz 2009).
To conclude, we think that the insights that QP brings to cognitive science, as reviewed by P&B, are likely to be as significant when
used for interpretation of brain network analysis. An examination of
the recent brain imaging literature suggests that such an approach
already is under way (Banerjee et al. 2012a; 2012b; Noppeney et al.
2004). Although the words used by functional brain imagers to
describe their methods differ from the language of QP, we have
attempted to point out that there are strong similarities in the fundamental mathematical problem each strives to solve.
ACKNOWLEDGMENT
This work was supported by the Intramural Research Program of the
National Institute on Deafness and Other Communications Disorders.
Uncertainty about the value of quantum
probability for cognitive modeling
doi:10.1017/S0140525X12002889
Christina Behme
Department of Philosophy, Dalhousie University, Halifax, NS B3H 4P9
Canada.
[email protected]
Abstract: I argue that the overly simplistic scenarios discussed by Pothos
& Busemeyer (P&B) establish at best that quantum probability theory
(QPT) is a logical possibility allowing distinct predictions from classical
probability theory (CPT). The article fails, however, to provide
convincing evidence for the proposal that QPT offers unique insights
regarding cognition and the nature of human rationality.
Pothos & Busemeyer (P&B) propose that many existing models
fail to account for all aspects of cognitive processing, and discuss
a number of phenomena for which the formal framework of classical probability theory (CPT) seems to fail. However, virtually all
examples were based on hypothetical reasoning or experiments
under artificial laboratory or Gallup poll conditions. Therefore,
even if quantum probability theory (QPT) can account better
for some of these phenomena, it is not clear that the much stronger claim that QPT offers unique insights regarding cognition and
the nature of human rationality is supported at this stage. To
support this stronger claim, P&B would have needed to establish
that the cases discussed are paradigm cases of human cognition
and not artifacts of the experimental setup. Further, to argue
for QPT, P&B need to do more than observe that QPT predictions
differ from CPT ones. Before arguing “that the quantum
approach to cognition embodies all the characteristics of good
theory” (sect. 6.4) P&B need to show in detail over a wide
range of cases that specific numerical QPT predictions are correct.
One rather serious problem is the informal and vague presentation throughout. There seem to be two components/aspects to
P&B’s QP model, distinguishing it from CP. One aspect is the
issue of and motivation for vectors space models, which could
be real but, nevertheless, would have associated non-classical
logical operators (disjunction, conjunction, negation, implication)
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
279
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
that could account for non-commutativity effects. The second
aspect is the issue of and motivation for complex vector spaces.
P&B attempt to justify their preference for QPT by stressing the
value of projection operators. But projection operators can be
defined on real (in fact, on any) vector space. Real vector spaces
are used in many areas such as machine learning and information
retrieval (e.g., van Rijsbergen, 2004; Widdows 2004). The key concepts required for similarity matching include inner product,
length, angle, and projection. But none of those are unique to
complex vector spaces. However, P&B discuss these notions in
the context of finite-dimensional complex vector spaces; their
Hilbert spaces. As the arguments seem to support only real
vector space models over classical set-theoretic models, P&B
would need an independent argument showing that similarity
matching for cognition requires complex numbers. P&B have not
shown that any of the examples require quantum probability amplitudes, which involve complex numbers. For example, the concept
of superposition does not require complex values but applies to
linear systems quite generally and hence to vector spaces. Conflating notions that need to be kept separate makes it difficult to evaluate whether the application of quantum probability is justified.
Turning to the work discussed, it appears that especially the
Linda case, on which much argumentative burden rests, is problematic for several reasons. First, the case is not based on an actual
person but on one that had been “constructed to be representative
of an active feminist and unrepresentative of a bank teller”
(Tversky & Kahneman 2004, p. 227). Reasoning under such artificial conditions seems hardly a prototypical case of rational
decision making. Further, not all participants committed the conjunction fallacy and the authors of the study point out that
“because individuals who have different knowledge or hold different beliefs must be allowed to assign different probabilities to the
same event no single value can be correct for all people” (Tversky
& Kahneman 2004, p. 221). It is likely that some participants in
such experiments rely, at least to some degree, on incorrect background information. This might explain the violation of the law of
total probability, the conjunction fallacy. Undoubtedly, these
experiments establish that models that assume fully rational
agents do not apply to humans. But they do little to indicate
whether QPT is a better model for human cognition than is
CPT. Possibly non-QPT models can account for the observed
anomalies if we assume that the reasoner relied to some degree
on incorrect information and/or did not reason fully rationally.
Similarly, the artificial conditions of the one-shot prisoner
dilemma seem to interfere in some (but not all) individuals with
rational decision making (Shafir & Tversky 2004). But more realistic reiterated prisoner dilemma tasks lead to different results
(e.g., Axelrod & Hamilton 1981; Nowak & Sigmund 1992).
These cases seem to establish that rationality breaks down in
some cases but not in others. This is very different from
quantum physics, in which it is always the case that “knowledge
of the position of a particle imposes uncertainty on its momentum” (sect. 1.1). Hence, the analogy between cognition and
quantum physics might be much weaker than P&B suggest.
They assume that any ad hoc models are inferior to a single,
uniform mathematical framework (whether it be CPT or QPT).
Given the complexity of human decision making, however, ad
hoc models that make better predictions might be preferable
over principled frameworks that make inferior predictions.
Another area of concern is the claim that “there are situations in
which the distinctive features of QP provide a more accurate and
elegant explanation for empirical data” (sect. 1.2). However, the
specific examples discussed involve only the interaction between
two parameters. In such a constrained setting, it may be true
that for quantum probability theory “the relevant mathematics is
simple and mostly based on geometry and linear algebra” (sect.
1.2). But it is not clear at all whether and how the proposed
models can account for the multiply intertwined interactions
between existing knowledge and any imaginable problem
solving task. Take the discussion of the interference effects
280
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
between “happiness” and “employment.” The superficial and
sketchy discussion of the superposition effects of just these two
variables seems to support the conclusion that emotional judgments are the results of constructive processes. However,
anyone accepting the claim that QPT “fundamentally requires a
constructive role for the process of disambiguating a superposition
state” (sect. 2.1), has to ask whether such disambiguation is possible in realistic cases in which all factors that affect the emotional
state of a person need to be considered. P&B seem to have no
principled way of determining whether two attributes are incompatible. But such a distinction would be required before we can
decide whether QPT is needed to replace CPT. The authors
observe that classical optimality “assumes a measurable, objective
reality and an omniscient observer” (sect. 5). This assumption is
problematic. But this has been discussed in the philosophical literature at least since Hume (1751/1999; see also Cartwright
1999), and current models do not attempt to make predictions
for all aspects of the “real, noisy, confusing, ever-changing,
chaotic world” (sect. 5). Many questions regarding the best
model for cognition remain open and it seems premature to
assert, “the QP cognitive program…is clearly an important direction for future research” (sect. 6.3, emphasis added).
ACKNOWLEDGMENTS
I thank David Johnson and Robert Levine for their detailed comments. All
remaining errors are mine.
The (virtual) conceptual necessity of quantum
probabilities in cognitive psychology
doi:10.1017/S0140525X12002890
Reinhard Blutnera and Peter beim Grabenb
a
ILLC, Universiteit van Amsterdam, Amsterdam, 1090 GE, The Netherlands;
Department German Language and Linguistics, Humboldt Universität zu
Berlin, 10099 Berlin, Germany.
[email protected]
[email protected]
http://www.blutner.de/ http://www.beimgraben.info/
b
Abstract: We propose a way in which Pothos & Busemeyer (P&B) could
strengthen their position. Taking a dynamic stance, we consider cognitive
tests as functions that transfer a given input state into the state after
testing. Under very general conditions, it can be shown that testable
properties in cognition form an orthomodular lattice. Gleason’s theorem
then yields the conceptual necessity of quantum probabilities (QP).
Pothos & Busemeyer (P&B) discuss quantum probabilities (QP)
as providing an alternative to classical probability (CP) for understanding cognition. In considerable detail, they point out several
phenomena that CP cannot explain, and they demonstrate how
QP can account for these phenomena. An obvious way to downplay this chain of arguments is by demonstrating that in addition
to CP and QP models, alternative approaches are possible that
could also describe the phenomena without using the strange
and demanding instrument of QP. For example, one could
argue that the conjunction puzzle can be resolved by simple heuristics (Gigerenzer 1997), and the question ordering effects can be
resolved by query theory (Johnson et al. 2007).
A general strategy to invalidate such criticism is to look for a
universal motivation of QP that is based on fundamental (architectural) properties of the area under investigation. As Kuhn (1996)
clarified, such basic assumptions constituting a theoretical paradigm normally cannot be justified empirically. Basic assumptions
that concern the general architecture of the theoretical system
(paradigm) are called design features. Using a term that is
common in the generative linguistic literature (Chomsky 1995;
2005), we will describe properties that are consequences of such
design features as applying with (virtual) conceptual necessity.
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
We believe that P&B could further strengthen their argument
by demonstrating that quantum probabilities are such a (virtual)
conceptual necessity. This can be achieved by adopting the
recent developments of “dynamification” in logic (van Benthem
2011) and cognition (Barsalou 2008) in which cognitive actions
play a much more significant role than do “static” propositions.
Among others (including Atmanspacher et al. 2002), Baltag and
Smets (2005) have given a complete axiomatization for quantum
action, based on the idea of a quantum transition systems. In
this view, the states of a system are identified with the actions
that can be performed on the states. In physics, the relevant
actions are measurements. In cognitive psychology, actions correspond to tests that subjects conduct (yes/no questions in the simplest case). Basically, a quantum dynamic frame is characterized
by a set of states Σ and a system T of subsets of Σ called testable
properties. Each testable property A ∈ T is characterized by a
unique (partial) transfer function PA, which describes how
testing of A changes an input state s ∈ Σ. The system of testable
properties and the corresponding transfer functions are characterized by some plausibility conditions. For example, the testable
properties are closed under arbitrary intersection, states are testable, testing a true property does not change the state [s ∈ A
implies PA(s) = s], and repeatability (P2A = PA). Further, there are
more technical axioms such as self-adjointness and a covering law.
Restricting testable properties in this way, Baltag and Smets
(2005) were able to prove (based on earlier work by Piron
[1976]) that quantum dynamic frames are isomorphic to the
lattice of the closed subspaces of a Hilbert space (with transfer
functions as projection operators). In the Baltag/Smets approach,
two states s and t are considered orthogonal if no measurement
can transfer s into t. Properties A and B are orthogonal if all
states of A are orthogonal to all states of B. If A and B are orthogonal, the corresponding subspaces of the Hilbert space are orthogonal as well (and vice versa). Mathematically, probabilities are
totally additive measure functions, in the classical case based on
Boolean algebras and in the quantum case based on orthomodular
lattices. The underlying algebra is decisive for the properties of
the resulting measure function. In the quantum case, Gleason’s
theorem states that the corresponding measure functions can be
expressed by the squared length of the projections of a given
state s (or more generally, as the convex closure of such functions;
for details, see the original article [Gleason 1957], and for a constructive proof see Richman & Bridges [1999]), that is, as QP.
Our view is further supported by P&B’s speculations about
implications for brain neurophysiology. In the algebraic approach,
even classical dynamic systems, such as neural networks, could
exhibit quantum-like properties in the case of coarse-graining
measurements, when testing a property cannot distinguish
between epistemically equivalent states. Beim Graben and Atmanspacher (2009) used this “epistemic quantization” for proving
the possibility of incompatible properties in classical dynamic
systems. In neuroscience, most measurements, such as electroencephalography or magnetic resonance imaging, are coarse-grainings in this sense. Therefore, the Baltag/Smets approach has
direct implications for brain neurophysiology, without needing to
refer to a “quantum brain” as indicated by P&B.
Summarizing, the Baltag/Smets approach provides an independent motivation of QP, which is not based on particular
phenomena but rather on independently motivated general conditions concerning the dynamics of testing. All the conditions
needed for the proof are formulated in purely dynamic terms.
This makes quantum dynamic frames especially appealing for
psychological approaches formulating operational cognitive
laws. Recent work by Busemeyer and Bruza (2012), Trueblood
and Busemeyer (2011), and Blutner (2012) is in this spirit.
Based on a dynamic picture of propositions and questions and
hence on the design principles of cognitive architecture, we
state that QP is a virtual conceptual necessity. Needless to say,
we regain CP (Kolmogorovian) by assuming that no test is changing the state being tested.
On the quantum principles of cognitive
learning
doi:10.1017/S0140525X12002919
Alexandre de Castro
Laboratório de Matemática Computacional – CNPTIA/Embrapa – Campinas,
13083-886 SP, Brazil.
[email protected]
http://www.cnptia.embrapa.br
Abstract: Pothos & Busemeyer’s (P&B’s) query about whether quantum
probability can provide a foundation for the cognitive modeling embodies
so many underlying implications that the subject is far from exhausted. In
this brief commentary, however, I suggest that the conceptual thresholds
of the meaningful learning give rise to a typical Boltzmann’s weighting
measure, which indicates a statistical verisimilitude of quantum behavior
in the human cognitive ensemble.
The principles of superposition and entanglement are central to
quantum physics. Quantum superposition is commonly considered to be a mapping of two bit states into one. Mathematically,
we can say that it is nothing more than a linear combination of
classical (pure) states. As to the quantum entanglement, it refers
to a short- or long-range operation in which a strongly correlated
state, a mixed state, is built from pure states. An important feature
of this mixed state is that it cannot be represented by a tensor
product of states, and once such an entangled system is constructed, it cannot be dissociated (Dirac 1999).
In their target article, Pothos & Busemeyer (P&B) elegantly
argue that there may be quantum principles – notably superposition and entanglement – at play in the context of human cognitive
behavior. They also draw attention to the pertinent idea that the
concept of quantum likelihood can provide a novel guidance in
cognitive modeling. Viewed in these terms, I tend to agree with
P&B because I, too, have identified both superposition and entanglement from the cognitive premises formulated within the
process of subsumption (assimilation) of information proposed
by Ausubel (1963; 1968).
From the point of view of the process of subsuming information,
the material meaningfully incorporated within an individual’s cognitive structure is never lost, but a process called “forgetting” takes
place in a much more spontaneous manner, because it is a continuation of the very process of associative subsumption by which one
learns. This forgetting state is an obliterative stage of the subsumption characterized by Ausubel as “memorial reduction to the least
common denominator” (Brown 2000). This “memorial reduction”
required for the acquisition of new meanings (knowledge) is
clearly (and remarkably) a conceptual process of quantum superposition of mental states, and, consequently, this cognitive behavior
can be generically expressed by a quantum operation of retention
of information, a cognitive squeeze, as follows: |bit〉 + |bit〉 = |
qubit〉. In addition, Ausubel (1963) claimed that, when the obliterative stage of subsumption begins, “specific items become progressively less dissociable as entities in their own right until they are no
longer available and are said to be forgotten.”
This “forgetting” theorized by Ausubel seems to reflect very well
the entanglement included in the central idea of quantum cognition
raised by P&B. In passing, Vitiello’s work (1995) quoted in the target
article also addressed the squeeze of information and the forgetting
dynamics to describe the cognitive behavior, although that work does
not properly refer to Ausubelian subsumption of information. Nevertheless, more in line with Ausubel’s premises, Brookes’ pioneering
work (1980) on the cognitive aspects in the information sciences
(Bawden 2008) provides a quantitative sharp bias of meaningful
learning, albeit one seldom examined from this perspective (Neill
1982; Cole 1997; 2011). Most of the works found in the literature
quote Brookes’ fundamental equation of information science, K(S)
+ I ⇄ K(S + ΔS) – here assumed as an obliterative synthesis that
exhibits short-term instability – as merely a shorthand description
of knowledge transformation, wherein the state of mind K(S)
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changes to another state K(S + ΔS), because of an input of information I, being Δ S as an indicator of the effect this transformation
(Bawden 2011; Cornelius 2002).
On the other hand, Brookes (1981) – although in an incipient
approach – conjectured outright that the recipient knowledge structure included in the fundamental equation could be quantitatively
treated, which in a subjacent manner, links his work to the assimilative schema of information expressed by the Ausubelian symbolic
quantities, even though these quantities are situated in a semiquantitative pictorial landscape (Novak 2011; Moreira 2011; Seel 2012).
In support of this idea, Todd (1999) also advocated that the unit of
information embedded in Brookes’ theory is a concept derived
from Ausubel’s learning theory. For such reasons, I am convinced
that Brookes’ equation faithfully shapes the Ausubelian retention
of information, or, more specifically, the superposition and entanglement of information underlying the subsumption.
Interestingly, if we take into account that information is the
boundary condition of the human cognitive system – and if we
continue to perceive knowledge from a Nietzschean viewpoint,
in which subject and object are confused – then the reciprocal
reckoning of Brookes’ equation, K(S + ΔS) → K(S) + I, in addition
to providing a typical scenario of information retention, also seems
to give us a symbolic (and quantum) translation of Jose Ortega y
Gasset’s famous maxim, “I am I plus my circumstances,” which
Gasset (1998) placed at the metaphysical core of his epistemological approach of perspectivism.
Brookes himself addressed a peculiar notion of perspectivism in
his work. In a pioneering way, Brookes (1981) suggested in a reductionist geometric context – although without clarification – a rough
sketch, a skeleton, of a logarithmic equation to represent the carrying
of information into the human mind on the same basis as Hartley’s
law (Seising 2010), seeing that Hartley’s law – predecessor to Shannon’s idea of channel capacity – had, up to that time, been designed
solely to handle information in a purely physical system. However,
albeit Brookes has made a valuable contribution by suggesting a
Hartley-like behavior for information processing in the mind, he
was not able to identify the appropriate cognitive variables for the
implementation of his physicalistic approach from the perspectivism.
I showed in a recent preprint (Castro 2013) that the conceptual
schema of meaningful learning leads directly to a ShannonHartley-like model (Gokhale 2004), and that this model can be
interpreted from basilar cognitive variables, such as information
and working memory. Moreover, starting this learning schema, I
have found that the ratio between the two mental states given by
the Brookes’ fundamental equation of information science is as
follows: K(S)/K(S + DS) ≥ e−DE/kB T , where ΔE is the free energy
of the ensemble, kB is Boltzmann’s constant, and T is the absolute
temperature. The so-called Boltzmann factor, e−DE/kB T , is a weighting measure that evaluates the relative probability of a determined
state occurring in a multi-state system (Carter 2001) – that is, it is
a “non-normalized probability” that needs to be “much greater”
than unity for the ensemble to be described for non-quantum statistics; otherwise, the system exhibits quantum behavior.
As a result, this calculation shows that the internalization of one unit
of information into an individual’s mental structure gives rise to a
Boltzmann “cognitive” factor ≤2, which provides us a circumstantial
evidence that the subsumption of new material, as a cognitive process,
requires a quantum-statistical treatment, such as P&B have proposed.
Cold and hot cognition: Quantum probability
theory and realistic psychological modeling
doi:10.1017/S0140525X12002907
Philip J. Corr
School of Psychology, University of East Anglia, Norwich Research Park,
Norwich NR4 7TJ, United Kingdom.
[email protected]
http://www.ueapsychology.net/
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Abstract: Typically, human decision making is emotionally “hot” and does
not conform to “cold” classical probability (CP) theory. As quantum
probability (QP) theory emphasises order, context, superimposition
states, and nonlinear dynamic effects, one of its major strengths may be
its power to unify formal modeling and realistic psychological theory
(e.g., information uncertainty, anxiety, and indecision, as seen in the
Prisoner’s Dilemma).
Classical probability (CP) theory has struggled to provide a comprehensive, sometimes even adequate, description and explanation of
human decision making. This conclusion has pervaded psychology
and related disciplines, for example in economics, where the
notion of a rational single-type homo economicus is fast falling out
of favour and being replaced by one that highlights the heterogeneity of economic agents and their decision- making processes. Many
behavioural scientists doubt that formal modeling based on CP principles can describe and explain even relatively simple laboratorybased behaviour, let alone the everyday examples that, typically,
entail some degree of emotional activation by virtue of the different
payoffs associated with different possible outcomes. The fact that
different outcomes exist is, itself, sufficient to induce conflictrelated emotion, thus rending typical human decision making “hot.”
Whatever the ultimate value of quantum probability (QP)
theory, it represents a promising way forward to model the computations of cognition involved in complex psychological situations. The need for such a formal modeling theory is
highlighted by the trend in recent years towards an integration
of “cold” cognition with “hot” (emotional) processes and decision
making, as seen in the “cognition-emotion” literature. CP theory is
poorly equipped to meet this challenge. This commentary
explores the potential of QP theory to provide a general modeling
approach for realistic psychology theory.
In the typical laboratory situation, the experimenter tries to
control all extraneous variables to isolate only those of interest;
however, in the case of human studies, internal states of the participant cannot be controlled. Participants bring to the experimental situation their own expectations, desires, and habitual modes of
thinking and behaving; and, worse still for the experimentalist,
these factors differ among individuals. This fact is seen readily
in experimental games studied by behavioural economists, in
which understanding of behaviour is impoverished by failing to
account for dispositional differences in preferences and personality (Ferguson et al. 2011).
The above-mentioned points can easily be illustrated in relation
to the “sure thing” principle seen with the prisoner’s dilemma
problem. The literature shows: (1) knowing that one’s partner
has defected leads to a higher probability of defection; (2)
knowing that one’s partner has cooperated also leads to a higher
probability of defection; and, most troubling for CP theory, (3)
not knowing one’s partner’s decision leads to a higher probability
of cooperation. Everyday equivalents of this situation do not, typically, resemble the decision-making dynamics assumed by CP
theory: people have psychological processes that it fails to
model, or to even consider to be relevant. For example, in this
situation, outcome (1) could simply be (self-serving) retaliation;
and in the case of (2) taking the easier route, but, we should
wonder, what would happen if the partner was a loved one
(e.g., your child)? Scenario (3) is a situation of uncertainty and
psychological conflict (at least as perceived by many participants),
and we know from the literature that such conflicts lead to cautious, risk-assessing behaviour which should be expected to
lower rates of defection (which represents a final decision
devoid of any cognitive dithering). Not knowing one’s partner’s
decision is, in psychological if not logical terms, very different to
outcomes (1) and (2); therefore, we should expect a different
decision outcome.
The main point of the previous discussion is that in a situation
of information uncertainty, people’s decisions will be influenced
by a host of factors, including evaluation of likely payoffs as well
as consideration of reputational damage and likely carry-over
effects to other situations. Most decision situations outside the
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
laboratory do not resemble the constraints of the one-shot
experiment, and in this situation people find it difficult to
break free from their habitual forms of cognition. In actual
fact, most laboratory-based decision-making situations are not
as tightly constrained as assumed by the experimenter; for
example, the experimenter typically knows the participant’s
decision and the participants know that the experimenter
knows. This is relevant information from the participant’s
point of view. In designing experimental situations, it is important to make things as simple as possible, but not to the extent
that the situation loses touch with psychological reality and,
therefore, external validity.
As an example of the consequences of information uncertainty, which is inherent in the prisoner’s dilemma problem,
psychological entropy has been used to account for its effects.
As noted by Hirsh et al. (2012), “As a system’s disorder and
uncertainty increases, its ability to perform useful work is hampered by reduced accuracy in specifying the current state, the
desired state, and the appropriate response for transforming
the former into the later” (p. 305). They further note that uncertainty leads to goal-conflict and anxiety, which adds further
emotional heat to the cognitive system. And, as with many
other processes, in the specific example of conflict-related
anxiety, we observe a change in the balance between controlled-reflective and automatic-reflexive behaviour (Corr
2011) which can, and often does, lead to nonlinear dynamic
effects of the type predicted by QP theory. In addition, the perception of potential rewards and punishments in the situation
trigger prepotent automatic reactions (e.g., in the case of goalconflict anxiety, behavioural inhibition and cognitive rumination;
McNaughton & Corr 2004) which we may well expect to impact
on decision-making processes in complex, but unpredictable,
ways. Therefore, CP modeling of the prisoner’s dilemma problem
does not tell us much about how people actually behave in “hot”
decision- making situations.
In contrast, one major opportunity of QP theory is the provision of a general computational framework for the modeling
of dynamic, and realistic, psychological processes. As Pothos
& Busemeyer (P&B) observe, QP theory has strong potential
in this regard because it takes account of processes that are
strongly order and context dependent, recognises that individual states are often superimposition states, and assumes that
composite systems are often entangled and cannot be composed into their separate subsystems. It would be valuable for
P&B to consider how their proposals for QP theory might be
extended to start to address emotionally hot cognition and
behaviour.
Beyond quantum probability: Another
formalism shared by quantum physics and
psychology
doi:10.1017/S0140525X12002920
Ehtibar N. Dzhafarova and Janne V. Kujalab
a
Department of Psychological Sciences, Purdue University, West Lafayette, IN
47907; bDepartment of Mathematical Information Technology, University of
Jyväskylä, Jyväskylä FI-40014, Finland.
[email protected]
jvk@iki.fi
http://www2.psych.purdue.edu/∼ehtibar http://users.jyu.fi/∼jvkujala
Abstract: There is another meeting place for quantum physics and
psychology, both within and outside of cognitive modeling. In physics it
is known as the issue of classical (probabilistic) determinism, and in
psychology it is known as the issue of selective influences. The
formalisms independently developed in the two areas for dealing with
these issues turn out to be identical, opening ways for mutually
beneficial interactions.
Pothos & Busemeyer (P&B) present a compelling case for
quantum formalisms in cognitive modeling. This commentary
is more of an addendum, about another area in which psychology meets quantum physics, this time as a result of the coincidence of formalisms independently developed and motivated.
In quantum physics, they grew from the investigation of the
(im)possibility of classical explanation for quantum entanglement, in psychology from the methodology of selective influences. Surprisingly, the meeting occurs entirely on classical
probabilistic grounds, involving (at least so far) no quantum
probability.
The issue of selective influences was introduced to psychology
in Sternberg’s (1969) article: the hypothesis that, for example,
stimulus encoding and response selection are accomplished by
different stages, with durations A and B, can be tested only in
conjunction with the hypothesis that a particular factor (experimental manipulation) α influences A and not B, and that some
other factor β influences B and not A. Townsend (1984) was
first to propose a formalization for the notion of selectively influenced process durations that are generally stochastically dependent. Townsend and Schweickert (1989) coined the term
“marginal selectivity” to designate the most conspicuous necessary condition for selectiveness under stochastic dependence: if
α→A and β→B, then the marginal distribution of A does not
depend on β, nor does the marginal distribution of B depend
on α. This condition was generalized to arbitrary sets of inputs
(factors) and outputs (response variables) in Dzhafarov (2003).
Selectiveness of influences, however, is a stronger property, as
the following example demonstrates. Let α, β be binary 0/1
inputs. Consider outputs
A = a + N, B = b + (1 − 2ab)N
where N is standard-normally distributed. The influence of α on
B is obvious, but B is distributed as Norm(mean=β, variance=1),
that is, marginal selectivity is satisfied.
It was first suggested in Dzhafarov (2003) that the selectiveness
of influences (α→A, β→B,…) means that A, B,… can be presented as, respectively, f (α, R), g(β, R),… where R is some
random variable and f, g,… are some functions. By now, we
know several classes of necessary conditions for selectiveness,
that is, ways of looking at joint distributions of A, B,… at different
values of inputs α, β,… and deciding that it is impossible to represent A, B,... as f (α, R), g(β, R),… (Dzhafarov & Kujala 2010;
2012b; in press b; Kujala & Dzhafarov 2008). For special classes
of inputs and outputs we also know conditions that are both
necessary and sufficient for such a representability. Thus, if A,
B,…, X and α, β,…, χ all have finite numbers of values, then
α→A, β→B,.., χ→X if and only if the following linear feasibility
test is satisfied: the linear system MQ = P has a solution Q with
non-negative components, where P is a vector of probabilities
for all combinations of outputs under all combinations of inputs,
and M is a Boolean matrix entirely determined by these combinations (Dzhafarov & Kujala 2012b).
In physics, the story also begins in the 1960s, when Bell (1964)
found a way to analyze on an abstract level the Bohmian version
of the Einstein–Podolsky–Rosen (EPR/B) paradigm. In the simplest case, the paradigm involves two spin-half entangled particles running away from each other. At some moment of time
(with respect to an inertial frame of reference) each particle’s
spin is being measured by a detector with a given “setting,”
which means a spatial orientation axis. For every axis chosen
(input) the spin measurement on a particle yields either “up”
or “down” (random output). Denoting these binary outputs A
for one particle and B for another, let the corresponding settings
be α and β. Special relativity prohibits any dependence of A on β
or of B on α. Bell formalized classical determinism with this prohibition as the representability of A, B as f (α, R), g(β, R), with the
same meaning of f, g, and R as above. Then he derived a necessary
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condition for such a representability, in the form of an inequality
involving three settings (x, y for α and y, z for β). The characterizations of Bell’s derivation as “one of the profound discoveries of the
[20th] century” (Aspect 1999) and even “the most profound discovery in science” (Stapp 1975) are often quoted in scientific and
popular literature. A generalization of this inequality to any
binary α and β, known as CHSH inequalities (after Clauser et al.
1969), was shown by Fine (1982) to be a necessary and sufficient
condition for representing A, B as f(α, R), g(β, R) (assuming marginal selectivity). CHSH inequalities are a special case of the
linear feasibility test mentioned previously as developed in psychology. In physics, this test is described in Werner and Wolf (2001a;
2001b) and Basoalto and Percival (2003).
How does one explain these parallels between the two very
different issues? The answer proposed in Dzhafarov and Kujala
(2012a; 2012b) is that measurements of noncommuting observables on one and the same particle are mutually exclusive,
because of which they can be viewed as different values of one
and the same input. Different inputs in the EPR/B paradigm
are spin measurements on different particles, whereas input
values are different settings for each particle. This is completely
analogous to, for example, α = left flash and β = right flash in a
double-detection experiment being inputs for two judgments
(A = I see/don’t see α, B = I see/don’t see β), and the intensities
of either flash being input values.
These parallels could be beneficial for both psychology and
physics. Thus, the cosphericity and distance tests developed in
psychology (Dzhafarov & Kujala, in press b; Kujala & Dzhafarov
2008) could be applicable to non-Bohmian versions of EPR, for
example, involving momentum and location. We see the main
challenge, however, in finding a principled way to quantify and
classify degrees and forms of both compliance with and violations
of selectiveness of influences (or classical determinism). We only
have one alternative to classical determinism in physics: quantum
mechanics. This may not be enough for biological and social behavior (Dzhafarov & Kujala, in press a).
Quantum probability and cognitive modeling:
Some cautions and a promising direction in
modeling physics learning
doi:10.1017/S0140525X12002932
Donald R. Franceschetti and Elizabeth Gire
Department of Physics and Institute for Intelligent Systems, The University of
Memphis, Memphis, TN 38152.
[email protected]
[email protected]
Abstract: Quantum probability theory offers a viable alternative to
classical probability, although there are some ambiguities inherent in
transferring the quantum formalism to a less determined realm. A
number of physicists are now looking at the applicability of quantum
ideas to the assessment of physics learning, an area particularly suited to
quantum probability ideas.
The notion that the theory of measurement and the logic of
quantum probability can be applied to mental states is almost as
old as quantum theory itself, and has drawn the attention of
such outstanding physicists and mathematicians as Bohr (who
adapted James’s notion of complementarity to physics [Pais
1991]) and Penrose (1989). Pothos & Busemeyer’s (P&B’s) analysis provides a cogent statement of the case for using quantum
probability. The prospect that quantum theory can be useful
outside the realm of atomic and subatomic physics is, or ought
to be, exciting to the community of physicists. We feel that we,
as physicists with a strong interest in cognitive aspects of physics
learning, can contribute to the discussion.
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In its simple form, quantum theory asserts that the maximally
determined states of a quantum system constitute a Hilbert
space, with certain rules for the evolution of the system in time
and for calculating the probabilities of various measurement outcomes. Hilbert spaces are generalizations of the ordinary two and
three-dimensional vector spaces one might have encountered in
high school or college. A Hilbert space is a collection of vectors
with a defined inner product that allows one to determine the
extent to which the vectors point in the same direction. If |ψ>
and |φ> are vectors in the Hilbert space, then we denote the
scalar product as <w | ψ> using the notation introduced by
Dirac (1958) and adopted by P&B. For atomic systems, |ψ> is a
complex quantity, vector, or function having real or imaginary
parts whose scalar product with itself equals one, <ψ|ψ> = 1. The
probability that a measurement of a property made on the system
in state |ψ> yields a measured value consistent with the state |w>
is given by the absolute square |<w|ψ>|2. In strict quantum
theory, once the measurement consistent with |ϕ>is made, the
system is fully in the state |w>, with <w|w> = 1. This re-normalization process is not apparent in the analysis of the target article. It is,
however, not needed if one considers only relative probabilities.
Measurable quantities in quantum theory are represented by
operators that transform the vectors of the Hilbert space. Measurable quantities A and B are compatible when the operators A and
B commute, that is, for all possible states |w>, AB|w>BA|w>. The
non-commutation of any A and B is ultimately determined by the
basic lack of commutation of position and momentum of a particle, which gives rise to the uncertainty principle for the operators
for position and momentum, which are conjugate operators in a
technical sense (Bohr’s complementarity applied to the most basic
mechanical measurements). When quantum probability is applied
to mental states or likelihood judgments, as is done by P&B, one
must make a number of assumptions about the predicates “is a feminist” or “is a bank teller,” which fall outside the realm of physics in
which complementary attributes are easier to define.
For systems not maximally determined, that is, systems that
have not passed through a sufficient number of filters to determine the values of a full set of commuting observables, a single
state vector in Hilbert space cannot provide a complete description. Such systems are more generally said to be in mixed states,
which are incoherent superpositions of the pure quantum states,
and are described by a density operator on the Hilbert space of
possible maximally specified states. In terms of probable outcomes of measurements, the mixed state allows interpolation
between the realm of Bayseian probability in which P(AB|A) is
P(A)P(B|A) and quantum probability, in which the Baysian
assumptions absolutely do not hold for sequential measurements
of incompatible measurements.
Treatment of mixed states is possible using the density matrix
formalism introduced by Von Neumann in 1927, which is concisely summarized by Fano (1957). The trace of the density
matrix (sum of its diagonal elements) is unity, as the system
must be in some state. Multiply the density matrix by itself,
however, and the trace is unity only if the system is in a pure,
that is, maximally characterized, state. For systems in a mixed
state, it is possible to define an entropy function that characterizes
our lack of knowledge about the state. Making measurements on a
system in a mixed state produces a mixed state of lower entropy or
a pure state.
An important illustration of the possibilities suggested by the
density matrix is provided by the work of Bao et al. (2002). In
studying the mental models used by classes of undergraduates
in answering a group of related physics questions, they assign a
vector of unit length to each student, depending on the models
used by the student, and then from a density matrix to represent
the entire class. As the class evolves with instruction, the density
matrix changes and approaches that for a pure state. As the students progress in their learning, the density matrix changes, as
testing is a form of measurement. Ultimately, one ends up with
a pure state for the well-instructed class.
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
Our comments are not intended to disparage the importation of
ideas from quantum probability to applications in psychology, but
rather to draw attention to the richness of the formalism and some
possible pitfalls. We hope that there will be continuing interaction
among quantum theorists, physics teachers, and psychologists, to
their mutual benefit.
Does quantum uncertainty have a place in
everyday applied statistics?
doi:10.1017/S0140525X12002944
Andrew Gelman and Michael Betancourt
Department of Statistics, Columbia University, New York, NY 10027.
[email protected]
[email protected]
http://www.stat.columbia.edu/∼gelman
Abstract: We are sympathetic to the general ideas presented in the article
by Pothos & Busemeyer (P&B): Heisenberg’s uncertainty principle seems
naturally relevant in the social and behavioral sciences, in which
measurements can affect the people being studied. We propose that the
best approach for developing quantum probability models in the social
and behavioral sciences is not by directly using the complex probabilityamplitude formulation proposed in the article, but rather, more
generally, to consider marginal probabilities that need not be averages
over conditionals.
We are sympathetic to the proposal of modeling joint probabilities using a framework more general than the standard model
(known as Boltzmann in physics, or Kolmogorov’s laws in probability theory) by relaxing the law of conditional probability, p
(x)=Σy p(x|y)p(y). This identity of total probability seems perfectly natural, but is violated in quantum physics, in which the
act of measurement affects what is being measured, and it is
well known that one cannot explain this behavior using the standard model and latent variables. (There have been some
attempts to reconcile quantum physics with classical probability,
but these resolutions are based on expanding the sample space so
that measurement is no longer represented by conditioning, thus
defeating the simplicity of the probabilistic approach.) The generalized probability theory suggested by quantum physics might
very well be relevant in the social sciences.
In standard probability theory, the whole idea of conditioning is
that there is a single joint distribution – parts of which may be unobserved or even unobservable, as in much of psychometrics – and that
this distribution can be treated as a fixed object measurable through
conditioning (e.g., the six blind men and the elephant). A theory that
allows the joint distribution to change with each measurement could
be appropriate for models of context in social science, such as Mischel’s idea of allowing personality traits to depend on the scenario.
Just as psychologists have found subadditivity and superadditivity
of probabilities in many contexts, we see the potential gain of thinking about violations of the conditional probability law. Some of our
own applied work involves political science and policy, often with
analysis of data from opinion polls, where there are clear issues of
the measurement affecting the outcome. In politics, “measurement”
includes not just survey questions but also campaign advertisements,
get-out-the-vote efforts, and news events.
We propose that the best way to use ideas of quantum uncertainty
in applied statistics (in psychometrics and elsewhere) is not by directly
using the complex probability-amplitude formulation proposed in the
article, but rather by considering marginal probabilities that need not
be averages over conditionals. In particular, we are skeptical of the
proposed application of quantum probability to the famous “Linda
example.” Kahneman and Tversky’s “representativeness heuristic” is
to us a more compelling model of that phenomenon.
How exactly would we apply a quantum probability theory to
social science? A logical first step would be to set up an experiment
sensitive to the violation of the law of conditional probability: a twoslit-like model for a social statistics setting in which measurement
affects the person or system being measured. Consider, for
example, a political survey in which the outcome of interest, x, is
a continuous measure of support for a candidate or political position, perhaps a 0–100 “feeling thermometer” response. An intermediate query, y, such as a positive or negative report on the
state of the economy, plays the role of a measurement in the Heisenberg sense. The marginal distribution of support might well be
different than the simple mixture of the two conditional distributions, and we would consequently expect p(x)≠Σy p(x|y)p(y).
A more sophisticated approach, and at the same time a stronger
test of the need for quantum probabilities, is akin to the original
Stern–Gerlach experiments. Participants would be asked a series
of polarizing questions and then split by their responses. Those
two groups would then be asked a further series of questions,
eventually returning to the initial question. If, after that intermediate series of questions, a significant number of participants
changed their answer, there would be immediate evidence of
the failing of classical probabilities, and a test bed for quantum
probability models.
The ultimate challenge in statistics is to solve applied problems.
Standard Boltzmann/Kolmogorov probability has allowed
researchers to make predictive and causal inferences in virtually
every aspect of quantitative cognitive and social science, as well
as to provide normative and descriptive insight into decision
making. If quantum probability can do the same – and we hope
it can – we expect this progress will be made as before: developing
and understanding models, one application at a time.
Cognitive architectures combine formal and
heuristic approaches
doi:10.1017/S0140525X12002956
Cleotilde Gonzaleza and Christian Lebiereb
a
Social and Decision Sciences Department, Dynamic Decision Making
Laboratory and bPsychology Department, Carnegie Mellon University,
Pittsburgh, PA 15213.
[email protected] [email protected]
http://www.cmu.edu/ddmlab/
Abstract: Quantum probability (QP) theory provides an alternative
account of empirical phenomena in decision making that classical
probability (CP) theory cannot explain. Cognitive architectures combine
probabilistic
mechanisms
with
symbolic
knowledge-based
representations (e.g., heuristics) to address effects that motivate QP.
They provide simple and natural explanations of these phenomena
based on general cognitive processes such as memory retrieval,
similarity-based partial matching, and associative learning.
Pothos & Busemeyer (P&B) must be lauded for providing an
alternative way to formalize probabilities in cognitive models in a
world where classical probability (CP) theory dominates modeling.
The findings that they discuss are indeed a challenge for CP. Existing
heuristic explanations are often unsatisfactory, offering few detailed
quantitative explanations of the cognitive processes involved. For
example, how do heuristics emerge and how do they relate to a
formal representation of psychological processes? P&B demonstrate
how quantum probability (QP) theory addresses these challenges.
Here, we argue that cognitive architectures, a modeling
approach with a long history in the cognitive sciences, may also
address the outlined challenges. The main article contained
little discussion of cognitive architectures, with the few examples
showcasing ones that rely heavily on CP. Whereas cognitive architectures do have a probabilistic aspect with stochastic components
to processes such as memory retrieval or action selection, they
combine probabilistic processing (i.e., CP) with symbolic knowledge-based representations (e.g., heuristics).
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Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
Figure 1 (Gonzalez & Lebiere).
Attributes and associations to the options in the Linda problem.
Cognitive architectures are computational implementations of
cognitive theories that unify and represent a full range of
human cognitive processes from perception to action (Newell
1990). Their strengths are derived from a tight integration of
their different components, particularly those satisfying the functional constraints to help maintain the “big picture” needed to
understand the human mind (Anderson & Lebiere 2003). Performance in any given task is a result of the complex interactions
between various modules, their underlying mechanisms, and the
resulting information flows. Adaptive Control of Thought—
Rational (ACT-R) (Anderson 2007; Anderson & Lebiere 1998)
is one of the most well-known architectures, with hundreds of
published models representing a broad range of tasks and
phenomena.1 Its distinguishing feature is the ability to combine
symbolic representations (declarative chunks and procedural
rules) with subsymbolic processes tuned by statistical learning.
This allows ACT-R to create symbolic representations of heuristics and knowledge in the form of production rules and declarative
chunks with probabilistic processes that can capture many mechanics of CP and even QP. In fact, there are researchers who have
taken up the task of explaining a portion of the large collection of
cognitive biases and heuristics through memory processes. These
studies often use ACT-R cognitive models (Marewski & Mehlhorn
2011; Marewski & Schooler 2011; Schooler & Hertwig 2005).
Although these are commendable efforts, the large variety of
cognitive biases cannot be all explained by one single comprehensive model. Rather, these researchers offer multiple models: one
for each type of heuristic (Marewski & Mehlhorn 2011). We have
made a similar observation regarding models of decisions from
experience, which are often task-specific and are developed to
account for only one variation of a given task (Gonzalez & Dutt
2011; Lejarraga et al. 2012). Therefore, a modeling methodology
that provides a unified approach to a variety of tasks by leveraging
the same architectural mechanisms is needed. For example, an
instance-based learning (IBL) model of repeated binary choice
offers a broad and robust unified explanation of human behavior
across multiple paradigms and variations of these tasks (Gonzalez
2013; Gonzalez & Dutt 2011; Lejarraga et al. 2012).
IBL models are process models that rely on ACT-R learning and
memory mechanisms, most notably those involving declarative representations and the activation mechanisms. Activation is a value
corresponding to each chunk in memory, learned to reflect its
usage statistics. It controls information retrieval from memory
and can be interpreted as the log odds of retrieval. The activation
mechanisms involve four main processes: base-level learning that
accounts for recency and frequency of information, similaritybased partial matching that represents the mismatches between a
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task’s situational attributes and those stored in memory, associative
learning that reflects the impact of contextual values and the associations between cues in different contexts, and a stochastic component that makes the retrieval process probabilistic.
As an example, consider the Linda problem used by P&B. The
conjunction fallacy is explained by IBL models as follows. Two
main options (i.e., instances) are considered in the decision
process: (1) Linda as a bank teller, and (2) Linda as a bank teller
and a feminist. Whereas a formal probability theory such as QP
and CP represent the latter as a logical conjunction, they are represented as independent instances in memory in a computational
theory such as ACT-R. The activation of these two instances is determined by the attributes in the problem’s description, and in the
memory relationships to “bank teller” and to “feminist,” as is captured by the associative learning mechanism governing the spreading
of activation. A scenario of the described attributes and their possible
associations to the two options is presented in Figure 1. Without
going into detail regarding the activation processes, the figure
shows that the number of attributes associated with option 1 (3) is
less than the number of associations (7) related to option 2. Therefore, option 2 has a higher probability of retrieval given a higher activation, and it would be selected more often. Although this is an
impossibility according to CP, our example illustrates how IBL
models and ACT-R more generally account for this effect, and
explains it in terms of simple memory processes. When eschewing
a formal probabilistic framework in favor of a computational
account, apparent impossibilities simply dissolve in light of the cognitive processes used to actually produce the decisions.
In general, we believe that the advantages of QP over CP can all
be addressed through the mechanisms in ACT-R models. Superposition, for example, can be addressed through the similarity metrics
used in the partial matching mechanism in the activation equation,
entanglement can be addressed through the contextual effects of
associative memory, and incompatibility can be addressed
through the recency effects in the base-level learning processes.
In conclusion, although we agree with P&B on the large body of
evidence that goes directly against CP principles, we do not
believe that an alternative probabilistic theory is essential to
understanding the intuition behind human judgment in such conditions. Cognitive architectures are able to address these challenges in a natural way that leverages the characteristics of
general cognitive processes such as memory retrieval.
NOTE
1. ACT-R code, publications, and models are available at http://act-r.
psy.cmu.edu.
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
Quantum probability and comparative
cognition
doi:10.1017/S0140525X12002968
Randolph C. Grace and Simon Kemp
Department of Psychology, University of Canterbury, Private Bag 4800,
Christchurch, New Zealand.
[email protected]
[email protected]
Abstract: Evolution would favor organisms that can make recurrent
decisions in accordance with classical probability (CP) theory, because
such choices would be optimal in the long run. This is illustrated by the
base-rate fallacy and probability matching, where nonhumans choose
optimally but humans do not. Quantum probability (QP) theory may be
able to account for these species differences in terms of orthogonal
versus nonorthogonal representations.
Pothos & Busemeyer (P&B) are to be congratulated for their
target article, which presents an exciting new paradigm for modeling cognition. We suggest that by including a comparative perspective grounded in scientific realism, quantum probability
(QP) theory may have even more promise as a framework for
describing cognition in humans and nonhumans alike. Moreover,
from this view, QP and classical probability (CP) theory are viewed
as complementary, not contradictory, frameworks.
Shepard (1994) argued that organisms that evolved in a physical
world would likely have internalized the constraints of that world.
If QP theory provides the formal basis for the physical description
of the world, then it should also provide the formal basis for the
description of cognitive processes within organisms that evolved
in that world. (It is interesting to consider why quantum mechanics is hard to understand, but we leave that for another day).
However, P&B seem tempted to abandon scientific realism – prematurely, in our view. In the context of discussing the implications
of QP for human rationality, they write: “The main problem with
classical optimality is that it assumes a measurable, objective
reality,” and later, “Quantum theory assumes no measurable
objective reality” (sect 5). It is understandable that P&B draw
clear distinctions between CP and QP models, but there is a
risk of losing sight of their complementary aspect.
If, as most believe, CP describes the workings of chance in the
real world, evolution would favor organisms that are able to make
decisions in accordance with CP principles, particularly in situations in which recurrent choices are possible, because such
decisions would produce the best outcome in the long run.
There is considerable evidence for the utility of optimality
models in describing nonhuman behavior, both in the laboratory
and natural ecology (e.g., optimal foraging theory [Charnov
1976; Koselj et al. 2011; Stephens & Krebs 1986]; ideal free distribution [Krivan et al. 2008; Sebastian-Gonzalez et al. 2010]; behavioral economics [Hursh 1984; Rachlin et al. 1976]). However,
it is important to acknowledge that some theorists have proposed
evolutionary accounts of apparently “irrational” human behavior.
For example, De Fraja (2009) produced a plausible account of
why conspicuous consumption might be evolutionarily adaptive,
and Kenrick et al. (2009) argue for the modularity of utility
across different domains (see also Robson & Samuelson 2011).
Notably, there are situations in which nonhumans behave optimally and consistent with CP, whereas humans do not. A prominent example is the base-rate fallacy, associated with Tversky
and Kahneman’s (1980) “taxi cab” problem. Here subjects are
asked to estimate the probability that a taxi involved in a hitand-run accident was blue or green. They are told that 85% of
the cabs in the city are green, but that a witness identified the
cab as blue, whose accuracy was later found to be 80% under comparable conditions. Typically humans estimate the probability of
the cab being blue as greater than 50%, thus placing too much
weight on the witness testimony; the correct answer (from
Bayes’ theorem) is 41%. By contrast, nonhumans (pigeons) did
not neglect base rates and responded optimally when tested in
an analogue task (Hartl & Fantino 1996). Another example in
which humans fail to respond optimally but nonhumans do not
is probability matching (Fantino & Esfandiari 2002).
These species differences can be accommodated within QP
theory by assuming that nonhumans construct orthogonal representations in situations with multiple sources of information,
leading to behavior consistent with CP and optimality, whereas
humans do not. This corresponds to the distinction between
“compatible” and “incompatible” questions in the target article.
From a comparative perspective, the question is why and under
what conditions would humans be more likely to construct nonorthogonal representations (“incompatible questions”).
Comparative research on visual categorization suggests that
differences in attention-related processes may provide a clue. In
the randomization procedure, subjects view circular discs with
sine-wave gratings that vary in frequency and orientation, and
assign each to one of two categories (Ashby & Gott 1988;
Maddox et al. 2003). Two types of tasks are commonly used:
one in which category exemplars are defined in terms of differences along one dimension, whereas the other dimension is irrelevant, and one in which both dimensions are relevant. The
former is termed a “rule-based” (RB) task because accurate performance can be described by a verbal rule, whereas the latter
is known as “information integration” (II) because correct
responding depends on both dimensions. With training, humans
can learn to respond correctly on the II task but when debriefed
cannot describe their performance in terms of verbal rule, and
typically say that they responded on the basis of a “gut feeling”
(Ashby & Maddox 2005).
The RB task is one in which selective attention facilitates performance, whereas the II task requires divided attention.
Humans and nonhuman primates are able to learn the RB task
faster and to a higher degree of asymptotic accuracy than the II
task (Smith et al. 2010). In contrast, pigeons show no difference
in learning rate or accuracy (Smith et al. 2011). Smith et al.
(2012) have argued that these species differences suggest that
the primate lineage may have evolved an increased capacity for
selective attention. In terms of QP theory, selective attention
may facilitate a shift in task representation. Pigeons may be constrained to use an orthogonal representation, in which both
dimensions are given equal weight, leading to similar performance
on both tasks, whereas the representation used by human and
nonhuman primates is more flexible and allows for faster and
more accurate learning in the RB task.
We agree that QP theory is an exciting new direction for modeling of human cognition, but as the previous examples suggest, it
may benefit from a comparative perspective. An enduring controversy in comparative cognition has been the question of differences in intelligence across species (Macphail 1987), including
whether human and nonhuman differences should be understood
in qualitative or quantitative terms (Roth & Dicke 2005). We are
intrigued by the possibility that QP theory may ultimately help to
resolve this question.
Quantum mathematical cognition requires
quantum brain biology: The “Orch OR” theory
doi:10.1017/S0140525X1200297X
Stuart R. Hameroff
Departments of Anesthesiology and Psychology, The University of Arizona,
University of Arizona Medical Center, Tucson, AZ 85724.
[email protected]
www.quantum-mind.org
Abstract: The “Orch OR” theory suggests that quantum computations in
brain neuronal dendritic-somatic microtubules regulate axonal firings to
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control conscious behavior. Within microtubule subunit proteins,
collective dipoles in arrays of contiguous amino acid electron clouds
enable “quantum channels” suitable for topological dipole “qubits” able
to physically represent cognitive values, for example, those portrayed by
Pothos & Busemeyer (P&B) as projections in abstract Hilbert space.
Pothos & Busemeyer (P&B) suggest that human cognition can
follow rules of quantum, rather than classical (Bayesian) probability mathematics. P&B are agnostic on whether such cognition
involves actual quantum activity, but quantum mathematical cognition logically requires quantum biology. Could functional
quantum biology occur in the brain?
The Penrose–Hameroff “Orch OR” theory suggests that cytoskeletal microtubules inside brain neurons perform quantum
computations that mediate conscious perceptions and control
conscious behavior (Hameroff 1998; 2006a; 2007; 2012; Hameroff
& Penrose 1996a; 1996b; Penrose & Hameroff 1995; 2011). In
Orch OR, synaptic inputs “orchestrate” quantum superpositioned
states (quantum bits, or “qubits,” existing simultaneously in
alternative states) in “tubulin” subunit proteins in microtubules
in dendrites and soma/cell bodies of “integrate-and-fire” brain
neurons (Fig. 1), microtubules in which memory may be
encoded (Craddock et al. 2012b). Dendritic-somatic tubulin
qubits entangle, interfere, and compute according to the Schrödinger equation during integration phases, then spontaneously
reduce, or collapse to classical tubulin states that influence membrane polarization and regulate axonal firings, controlling behavior. (The Orch OR neuron is the “Hodgkin–Huxley” neuron
with finer scale, deeper-order microtubule quantum influences.)
At the end of each integration phase, quantum state reduction
occurs by an objective threshold (Penrose “objective reduction,”
“OR,” hence “orchestrated objective reduction”: “Orch OR”) for
example, every 25 ms, coinciding with “40 Hz” gamma synchrony
electro-encephalography (“EEG”). Orch OR is testable, feasible,
and, although criticized (McKemmish et al. 2009; Tegmark
2000; c.f. Hagan et al. 2001; Penrose & Hameroff 2011), is
increasingly supported by experimental evidence from quantum
biology (Engel et al. 2007; Gauger et al. 2011; Sarovar et al.
2010; Scholes 2010).
P&B take an extremely important step, but they omit four
important features, the first being consciousness. Quantum probability and quantum biology (e.g., Orch OR) may specifically
govern conscious cognition, whereas classical (Bayesian) probability and biology govern unconscious cognition (e.g., Hameroff
2010).
The second omission is information representation in the brain.
In a bold and novel assertion, P&B introduce projection rays in
abstract Hilbert space to represent cognitive values, for
example, feelings, or subjective “qualia,” such as “happy” and
“unhappy.” Could abstract quantum projections correlate with
actual quantum physical processes in brain biology? Are there
qubits in the brain?
According to Orch OR, collective van der Waals London force
dipoles among electron clouds of amino acid phenyl and indole
rings aligned in non-polar “quantum channels” within each
tubulin can orient in one direction or another, and exist in
quantum superposition of both directions, acting as bits and
qubits (Fig. 2a–d). Intra-tubulin channels appear to link to those
in neighboring tubulins to form helical quantum channels extending through microtubules (Fig. 3a). Anesthetic gas molecules that
selectively erase consciousness (sparing unconscious brain activity)
bind in these channels (Craddock et al. 2012a; Hameroff 2006b).
Collective dipole qubits in microtubule quantum channels can represent cognitive values, for example, P&B’s “happy/unhappy” and
“job/no job” (Fig. 3b).
Figure 1 (Hameroff). A neuron, and portions of other neurons are shown schematically with internal microtubules. In dendrites and cell
body/soma (left), microtubules are interrupted and of mixed polarity, interconnected by microtubule-associated proteins (MAPs) in
recursive networks (upper circle, right). Microtubules in axons are unipolar and continuous. Gap junctions synchronize dendritic
membranes, and may enable entanglement among microtubules in adjacent neurons (lower circle right). In Orch OR, microtubule
quantum computations occur during dendritic/somatic integration, and the selected results regulate axonal firings.
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Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
Figure 2 (Hameroff). a. A microtubule (A-lattice) is composed of peanut-shaped tubulin proteins. b. Within the microtubule, an
individual tubulin shows internal (colored) van der Waals radii of aromatic amino acids tryptophan (blue), phenylalanine (purple), and
tyrosine (green) traversing the protein (image provided with permission from Travis Craddock). c. Schematic tubulin illustrates
aromatic rings (phenyl shown here, although indole rings are also involved) arrayed through tubulin along “quantum channels.”
d. Collective dipoles along such channels may exist in alternative orientations, or quantum superposition of both orientations – that is,
quantum bits or “qubits.”
Collective dipoles in helical quantum channels appear suitable
for qubits, or “braids” in topological quantum computing in which
specific pathways, rather than individual subunit states, are fundamental information units, mediated by fractional charge “anyons”
(Collins 2006; Freedman et al. 2002; Kitaev 2003). Topological
qubits (or comparable “adiabatic” qubits) are inherently stable,
resistant to decoherence, and consistent with Orch OR (Hameroff
et al. 2002; Penrose & Hameroff 2011).
P&B’s third omission is quantum computation. They cite three
quantum features: (1) superposition (co-existing possibilities), (2)
entanglement (connectedness of separated states), and (3) noncommutativity (context-dependence, “incompatibility”). The
three features together comprise quantum computation in
which superpositioned “qubits” interfere and compute by entanglement, and then reduce/collapse to definite output states. In a
series of quantum computations, output of each reduction
Figure 3 (Hameroff). a. Seven tubulins in a microtubule A-lattice neighborhood are shown in which contiguous arrays of aromatic
amino acid rings traverse tubulin and align with those in neighboring tubulins, resulting in helical “quantum channel” pathways
throughout the microtubule. b. Dipole orientations in two different helical pathways along quantum channels represent alternate
cognitive values and their superpositions (as described by P&B, “happy/unhappy,” “job/no job”) – that is, topological dipole qubits.
c. Superpositioned qubits in (a larger portion of) a single A-lattice microtubule are shown during the integration phase, prior to
reduction/self-collapse by the Orch OR mechanism. d. Post-reduction values selected in the Orch OR process are illustrated, a
moment of conscious awareness having occurred.
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Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
provides input for the next, hence context-dependent incompatibility. P&B’s three features add up to cognitive quantum computation, as suggested in Orch OR.
The fourth omission, (and essence of the “measurement
problem” in quantum mechanics) is reduction of quantum superpositions to classical output states (“collapse of the wave function”), what P&B refer to as “disambiguating a superposition
state.” There are numerous theoretical suggestions as to how
and why superpositions reduce to classical states (multiple
worlds, decoherence, observer effect), but none address the
nature of superposition, and each is unsatisfactory for various
reasons (e.g., the von Neumann/Wigner “observer effect” puts
conscious observation outside science). Addressing this problem,
Penrose (1989; 1994; 1996; 2004) proposed (1) superpositions
were actual separations in underlying spacetime geometry, the
fine structure of the universe, (2) such separations were unstable,
and reduce to definite states after time t by an objective threshold
given by the indeterminacy principle E = −h /t (where E is the magnitude of the superposition/separation and −h is Planck’s constant
over 2π), and (3) such events result in moments of conscious experience and choice. Orch OR is the neurobiological framework for
Penrose OR, and perfect solution for P&B’s required “constructive
role for the process of disambiguating a superposition state.”
Figure 3c shows two superpositioned dipole pathways acting as
topological qubits in a microtubule A-lattice and representing
P&B’s “happy/unhappy” and “job/no job” (perhaps more properly
“job satisfaction” and “job dissatisfaction”). The qubits are
coupled; for example, “happy” and “job” are more likely to
coincide than “happy” and “no job”. During the integration
phase, superpositioned qubits entangle, evolve, and compute (in
a microtubule memory bed) by the Schrödinger equation until
threshold by E = −h /t is met, OR occurs, and classical microtubule
states are selected (correlating with “happy” and “job” in the
example given). According to Orch OR, consciousness occurs in
the end-integration moment of reduction, and microtubule
states selected in the OR process govern axonal firing and behavior (Hameroff 2012).
If cognition (and consciousness) utilize quantum mathematics,
entanglement, and non-commutativity, as P&B suggest, the brain
is likely to be using quantum computation, such as described in
Orch OR. From a well-known example of inductive reasoning:
“If the brain swims, looks, and quacks like a (quantum) duck,
then it probably is a (quantum) duck.”
ACKNOWLEDGMENTS
I thank Dave Cantrell for artwork, Travis Craddock for Figure 2b,
Hartmut Neven and Saatviki Gupta for comments, and Sir Roger
Penrose for collaboration.
Consider people’s tendency to overestimate conjunctive probabilities in the famous Linda case (Tversky & Kahneman 1983).
The QP model (P&B, Fig. 2) explains the conjunction fallacy as
follows. By projecting the initial state vector (Linda) first onto
the “feminist” vector, the angle with the “bank teller” vector is
thus reduced, and, therefore, the projection to “bank teller” via
“feminist” is stronger than the direct projection straight to
“bank teller.” To explain this effect psychologically, they then
offer the following suggestion: “once participants think of Linda
in more general terms as a feminist, they are more able to appreciate that feminists can have all sorts of professions, including being
bank tellers.” In some way, Linda is more easily imagined –
perhaps she seems more real – so that being considered a feminist
makes it more likely that she might also be a bank teller.
This is a totally novel explanation for this effect. Remember that
the initial story about Linda (that she was a liberal in college)
makes it very likely that she is a feminist. It is unclear why consideration of her being a feminist should then increase the subjective likelihood of her having an improbable job for a graduate such
as being a bank teller. This explanation bears no relation to the traditional use of representativeness to account for the conjunction
fallacy.
The conjunction fallacy bears close resemblance to another
phenomenon, also mentioned by P&B: the overextension of category conjunctions (Hampton 1987; 1988b). For example,
people are more likely to categorize chess as a “sport that is a
game” than as simply a “sport,” in much the same way that they
judge Linda as more likely to be a feminist bank teller than
simply a bank teller. The effect occurs across a range of different
categories (for example a ski-lift is more likely to be considered a
“machine that is a vehicle” than it is to be considered a vehicle, or
a refrigerator is more likely to be a “household appliance that is
furniture” than it is to be simply “furniture.”)
The composite prototype model (Hampton 1987; 1988b) developed to handle such cases proposes that the two conjuncts (for
example, feminist and bank teller) are composed into a single
composite concept, by merging their intensions (their associated
descriptive properties). Therefore, a sport that is a game will
have properties inherited from sports (competitive, skilled, and
physical) and also from games (rules and scoring). Further elaboration of the representation may then take place to improve the
coherence of the new composite (Hampton 1997). In terms of
Figure 2 in P&B’s article, a new vector would be formed for the
conjunction that roughly bisects the “feminist” and “bank teller”
vectors, as in Figure 1 here. As a consequence, provided the
Quantum probability and conceptual
combination in conjunctions
doi:10.1017/S0140525X12002981
James A. Hampton
Department of Psychology, City University London, City University,
Northampton Square, London EC1V OHB, United Kingdom.
[email protected]
www.staff.city.ac.uk/hampton
Abstract: I consider the general problem of category conjunctions in the
light of Pothos & Busemeyer (P&B)’s quantum probability (QP) account of
the conjunction fallacy. I argue that their account as presented cannot
capture the “guppy effect” – the case in which a class is a better member
of a conjunction A^B than it is of either A or B alone.
My commentary will focus on a specific point in the target article,
relating to the formation of conjunctions, either of subjective
probabilities or of degrees of category membership.
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Figure 1 (Hampton). Construction of a new vector for feminist
bank teller as intermediate between the two conjuncts.
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
initial state vector lies outside the angle formed by the two constituent concepts (as here), the projection of the initial state
vector onto the composite (conjunction) vector will always lie
between the projections on the two other vectors.
In psychological terms, Linda has many properties in favour of
her being a feminist, and hardly any in favour of her being a bank
teller; therefore, she has a moderate number of properties
suggesting that she is a feminist bank teller. The two degrees of
similarity are averaged. According to the representativeness heuristic, subjective probability is assessed through similarity (number
of relevant properties). Although the mechanism proposed by the
representativeness heuristic is very different from that proposed
by P&B, so far the predictions are similar. In terms of subjective
likelihood, Linda should most likely be a feminist, next a feminist
bank teller, and least likely a bank teller.
However, the two accounts can be dissociated. A critical test for
the QP account concerns the so-called “guppy effect.” In categorization tasks, people will sometimes claim that something is a
better example of a conjunction than of either of the conjuncts.
In their famous critique of the application of fuzzy logic to conceptual combination, Osherson and Smith (1981) pointed out that
guppies could be considered better pet fish than they were
either pets or fish. The account offered by P&B for conjunctions
will have trouble with this result. If the initial state vector must
first be projected to one vector and then onto the second, the
probability must always be reduced each time (see Fig. 2).
Therefore, it is not possible within this model to obtain this
double overextension of conjunctions. In contrast, the intensional
account offered by the composite prototype model does allow that
an example can be more similar to the composite than to either
conjunct. Pet fish lack some key properties of pets (they are not
warm and affectionate) and also lack typical properties of fish
(they are not caught in nets and fried with breadcrumbs). It is
therefore quite feasible for an exemplar such as the guppy,
which also lacks exactly these properties, to be a better fit to
the conjunction than to either conjunct. Although much rarer
than simple overextension of one conjunct, empirical data do in
fact support the guppy effect (Storms et al. 2005.) A better
vector representation would therefore be that shown in
Figure 3, where a new vector is created to represent the composite prototype of a pet fish, and the guppy projects more strongly
Figure 3 (Hampton). Projection from guppy to a new vector
representing the combination pet fish allows it to be a better pet
fish than it is either a pet or a fish.
onto this conjunctive vector than it does onto either the pet or the
fish vectors.
In conclusion, as a psychological account of conceptual combination, QP appears to lack sufficient resources to handle the intensional reasoning that can be seen to characterize much of human
thought (Hampton 2012). The notion of concepts as regions of
subspace makes good sense, as it captures the vagueness and
context-dependence of how concepts enter thoughts. However,
the simple projection mechanism suggested in Figure 2 is unlikely
to provide a rich enough framework to capture the ways in which
concepts combine.
Is quantum probability rational?
doi:10.1017/S0140525X12002993
Alasdair I. Houstona and Karoline Wiesnerb
a
School of Biological Sciences, University of Bristol, Bristol BS8 1UG, United
Kingdom; bSchool of Mathematics, University of Bristol, Bristol BS8 1TW,
United Kingdom.
[email protected]
[email protected]
http://www.bristol.ac.uk/biology/people/alasdair-i-houston/overview.html
http://www.maths.bristol.ac.uk/∼enxkw/
Abstract: We concentrate on two aspects of the article by Pothos &
Busemeyer (P&B): the relationship between classical and quantum
probability and quantum probability as a basis for rational decisions. We
argue that the mathematical relationship between classical and quantum
probability is not quite what the authors claim. Furthermore, it might
be premature to regard quantum probability as the best practical
rational scheme for decision making.
Figure 2 (Hampton). The P&B projection from guppy to fish,
and then to pet. The conjunction can never be greater than
both conjuncts.
Pothos & Busemeyer (P&B) provide an intriguing overview of
quantum probability as a framework for modeling decision
making. In this commentary, we concentrate on two aspects of
their article: the relationship between classical probability (CP)
and quantum probability (QP) and its implications for ideas about
rationality in humans and other animals.
In order to evaluate the contribution that QP makes to our
understanding of cognitive processes, it is necessary to be clear
about the relationship between CP and QP. In Section 4.3,
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Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
P&B reject the view that QP provides a better account of human
decision making “simply because it is more flexible.” We are not
completely happy with some of the statements that P&B make
about this issue. From a mathematical point of view, CP is
embedded as a special case in the more general non-commutative
(also referred to as “quantum”) probability theory. There are no
cases of CP distributions and the evolution thereof that cannot
be represented in a quantum probabilistic framework. Once the
Heisenberg matrix mechanics is used (equivalent to the Schrödinger wave vector notation used by P&B) it can be shown that any
commutative quantum theory can be written as a CP theory,
and vice versa, CP theory can be written as a commutative
quantum theory. Hence, CP is strictly a special case of non-commutative (quantum) probability (see Streater 2000). To clarify the
connection to P&B by example: what they call the “law” of double
stochasticity is merely a special case in quantum theory, that of
projective measurement, which is a subset of all (generalised)
measurements.
The discussion about “QP theory violating the law of total
probability” would be helped by clarifying the law of total probability. It states that the probabilities of all elements in a set
sum up to the probability of the whole set. In quantum mechanics
the whole can be less than the sum of its parts because of superposition of states. An equation such as the one in the target
article’s appendix
Prob(happy, unknown employment)
1
1
= M · U(t) · √ |cemployed l + M · U(t) · √ |cemployed l2
2
2
is more helpful if stripped down to the “superposition part” (the
state is “separable”), that is, the only information that matters is
Prob(unknown employment) = Prob(not employed)
+ Prob(employed)
Our final topic is the relationship between rational decision
making and natural selection. In this context, it is worth noting
that the decisions of nonhuman animals violate the principles of
rational decision making (Houston et al. 2007b). Given that
natural selection is expected to favour optimal choices, there is
a tension between natural selection and the observed behaviour
of animals. One response is to argue that animals only need to
make approximately optimal decisions in particular circumstances – the circumstances in which they have evolved. This
idea is related to the suggestion that decision making is based
on heuristics, as mentioned by P&B in Section 4.2. A more interesting possibility is that what appear to be suboptimal choices are
part of an optimal strategy for an environment that is richer than
was originally envisaged. For example, Houston et al. (2007a)
show that apparent violations of transitivity can be generated by
the optimal state dependent strategy given that the options presented to the animal will persist into the future. In essence, the
violations appear because the experimenter does not have the
same view of the world as the animal. Another example is provided by the work of McNamara et al. (2012) on state-dependent
valuation.
With these examples, we are trying to illustrate that the macroscopic world of decisions is more complex than traditional models
of decision theory assume. P&B (sect. 5) write: “If we cannot
assume an objective reality and an omniscient cognitive agent,
then perhaps the perspective-driven probabilistic evaluation in
quantum theory is the best practical rational scheme. In other
words, quantum inference is optimal, for when it is impossible
to assign probabilities to all relevant possibilities and combinations
concurrently.” We are not convinced that understanding the
macroscopic world of decisions requires a step as dramatic as
that of abandoning objective reality in the way that the quantum
world does.
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Limitations of the Dirac formalism as a
descriptive framework for cognition
doi:10.1017/S0140525X12003007
Artem Kaznatcheeva,b and Thomas R. Shultza,b
a
School of Computer Science; bDepartment of Psychology, McGill University,
Montreal, QC H3A 1B1, Canada.
[email protected]
[email protected]
http://www.cs.mcgill.ca/∼akazna/
http://www.tomshultz.net/
Abstract: We highlight methodological and theoretical limitations of the
authors’ Dirac formalism and suggest the von Neumann open systems
approach as a resolution. The open systems framework is a generalization
of classical probability and we hope it will allow cognitive scientists to
extend quantum probability from perception, categorization, memory,
decision making, and similarity judgments to phenomena in learning and
development.
Pothos & Busemeyer (P&B) present the Dirac formalism of
quantum probability (DQP) as a potential direction for cognitive
modeling. It is important to stress that the quantum probability
(QP) approach is a modeling framework. It does not provide guidance in designing experiments or generating testable predictions.
It is not a theory. P&B do not show how the framework could be
used to build predictive theories: all the examples listed are post
hoc descriptive models. Therefore, the target article should be
judged on its merits as a descriptive framework.
A framework has to allow experimenters to average behavior over
(or within) participants. Unfortunately, the pure states of P&B’s
Dirac formalism cannot be directly averaged. Experimental psychologists have addressed this methodological constraint of DQP by ad
hoc fixes (Bruza et al. 2009; Franco 2009). To avoid such fixes or
creating statistics outside the framework, one has to introduce
mixed states (see sect. 2.4 of Nielsen & Chuang [2000]), which physicists use to reason about open quantum systems.
A theoretical expressiveness constraint is that the time evolution
in DQP is inherently periodic. After a recurrence time, the evolution
will be arbitrary close to (experimentally indistinguishable from) the
identity, as a consequence of the Poincaré recurrence theorem for
closed quantum systems. In terms of decision making, this means
that given enough deliberation time, a participant will always
return to a mental state indistinguishable from the one before deliberation. As the QP framework has no natural time scales, this issue
can be swept under the rug by hoping that the recurrence time is
astronomical compared with typical deliberation. To overcome
this limitation in a principled manner, we must look beyond the
Dirac formalism and allow irreversible evolutions.
In quantum information theory, irreversible evolutions are
introduced via quantum channels (see sect. 8.2 of Nielsen &
Chuang [2000]). Like the mixed states, and unlike P&B’s pure
states and unitary evolution, quantum channels can be averaged;
a methodological asset. The combination of quantum channels
and mixed states is often described as the “von Neumann formalism” (NQP; or “Lindbland form” if we think in terms of master
equations) and is the standard approach for dealing with open
quantum systems. In physics, it is possible to alternate between
NQP and DQP by explicitly modeling the environment to close
the system. However, this is an unreasonable constraint for modeling an open system such as the mind. When modeling the
decision “Is Linda a bank teller or feminist?” psychologists do
not close the system by explicitly considering all the other concurrent thoughts (e.g., “What will I have for dinner?”).
Modellers must choose between NQP and DQP. The preceding
methodological and theoretical considerations compel us to select
NQP. Because a Markov chain is a type of quantum channel, this
choice makes the framework a generalization of classical probability
(CP; see Kaznatcheev [submitted] for a complete treatment) and
renders section 4.3 of the target article moot. We cannot use
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
global properties of the transition operator (such as the law of totalprobability versus double-stochasticity) to empirically decide
between CP and NQP, because any transition matrix consistent
with CP will also be consistent with NQP.
If we take NQP as a generalization of CP, then we are faced
with the difficulty of justifying the increase in complexity. Physicists are able to justify the difficulty of quantum mechanics over
classical local hidden variable theories through experimental violations of Bell’s inequality. Physical tests of Bell’s inequality rely on
being able to separate entangled subsystems (local property of the
transition operator) far enough and perform measurements fast
enough that the subsystems cannot communicate without violations of special relativity. Although Aerts and Sozzo (2011b) are
able to produce joint probability functions that violate Bell’s
inequality in psychological settings, they cannot draw the strong
conclusions of physicists because their experiments cannot
create a space-like separation between the subsystems (in their
case the systems are concepts within a single mind). In this
psychological setting, local and global hidden variable theories
cannot be distinguished, and a classical explanation cannot be
ruled out. Without an operationalist test (such as Bell’s) or reductionist grounding in the physics of quantum systems (which seems
unlikely [Litt et al. 2006]), we are left with only practical considerations as criteria to decide between the CP and NQP frameworks.
Because QP is notorious for being unintuitive and difficult to
understand, great care is required when trying to reconcile QP
with classical intuition. For example, non-orthogonal states are
impossible to perfectly distinguish (Chefles 2000). P&B make
this error in the classical reasoning of their model in section 3.1:
[I]n situations such as this, the more probable possible outcome
is evaluated first…Therefore, the conjunctive statement
involves first projecting onto the feminist basis vector, and subsequently projecting on the [non-orthogonal] bank teller one.
(emphasis ours)
Unfortunately, the subject cannot determine the more overlapped
state by any process inside QP. P&B have to step outside their framework and provide an ad hoc fix of the sort they dismiss in CP.
From a psychological perspective, it is encouraging that the QP
framework addresses issues in perception, categorization, memory,
decision making, and similarity judgments. However, it lacks tools
to cover learning and development. In contrast, Bayesian CP is
capable of learning and inference (Griffiths et al. 2008), and has
made inroads into modeling development (Shultz 2007). Comparable extensions of the QP framework would be welcomed. If QP
cannot be extended to learning and development, then it is unlikely
to replace the popular Bayesian CP approach.
In conclusion, we agree that a QP approach should be pursued in
psychology, but we suggest viewing it from the open systems perspective of NQP. DQP is restricted to closed systems, but the
mind is an open system, and, therefore, NQP yields a better modeling framework. However, it is unclear how to experimentally rule out
ad hoc CP interpretations without physically separating psychological
subsystems. P&B’s survey of QP’s significant progress in covering a
range of psychological phenomena is promising, and we look forward
to future coverage of learning and development.
Disentangling the order effect from the context
effect: Analogies, homologies, and quantum
probability
doi:10.1017/S0140525X12003159
Elias L. Khalil
Department of Economics, Monash University, Clayton, Victoria 3800,
Australia.
[email protected]
http://eliaskhalil.com
Abstract: Although the quantum probability (QP) can be useful to model
the context effect, it is not relevant to the order effect, conjunction fallacy,
and other related biases. Although the issue of potentiality, which is the
intuition behind QP, is involved in the context effect, it is not involved
in the other biases.
Why do we need quantum probability (QP) in place of classical
probability (CP)? According to Pothos & Busemeyer (P&B), we
need the QP technique to model “incompatible” questions, that
is, when the states of the world are “potential” states that would
collapse into concrete ones in light of interaction with other variables. This is similar to the famous Schrödinger’s cat: The state of
the cat is both dead and alive in the potential state, which becomes
concrete only with further interaction. But if the questions are
“compatible,” the CP technique would suffice.
P&B apply QP to the “context effect” as illustrated in the questions about happiness and employment. These questions are
incompatible because the evaluation of one interferes with the
evaluation of the other.
P&B, in turn, apply QP to the “order effect” and other biases such
as the conjunction fallacy, similarity judgment, and Wason selection
task. In these biases, incompatibility again arises because a particular
appearance of a question influences the answer.
But is a particular appearance – as in the “order effect” – the
same as the “context effect”? The order effect and other biases
definitely share similarities with the context effect – which seem
to justify the use of the same technique, namely, the QP technique
in their case.
However, there is problem: What if the similarities between the
appearance effect, such as the order effect, and the context effect
are analogies rather than homologies, to use the familiar biological
distinction? If they are analogies, the use of the same technique
(whether CP or QP) for the analysis of both effects would amount
to an “identificational slip” (Khalil 2000). One would commit an
identificational slip if one mistakes an analogy for a homology –
such as identifying the forearms of dolphins as “fins” or the forearms
of bats as “wings” (Khalil 2000). To recall, the forearms of dolphins
and bats are homologous because of common mammalian origin. In
contrast, the forearms of dolphins and fish, or the forearms of bats
and birds, are analogous because of common functions. One
would commit an identification slip if one used the fin or the wing
technique to describe how, respectively, dolphins swim or how
bats fly. Therefore, do P&B commit an identificational slip when
they use the QP technique to describe the order effect (and other
biases)?
First, what is the initial intuition behind QP? It is designed to
model potentiality in nature, with implications with regard to
the specification of meaning of what is observed by the sense,
including the meaning of observed actions in relation to goals.
The latter has ramifications for the understanding of internal
motivation, self-esteem, and entrepreneurship (see Khalil 1997a;
1997b; 2010). Such potentiality is a “context,” which becomes concrete with interaction.
Do the biases marshaled by P&B as candidates for QP involve
potentiality; a potentiality found in the context effect? With respect
to the order effect, do we have a potentiality issue?
When subjects are presented with A and B – for example, “Is
Clinton honest?” and “Is Gore honest?” – they give different
answers, depending on the order of A and B. The questions A and
B invoke symmetrical and independent information. Information
with regard to each question can be evaluated without the appeal
to the other. But this is not the case with the employment and happiness questions. If you ask if one is happy, the evaluation must depend
on another question, such as employment, because one must be
happy about something. But if you ask “Is Clinton honest?” the
evaluation need not depend on the honesty of Gore. Insofar as the
Gore question exercises influence on how to assess Clinton (i.e.,
the order effect), it must have acted as a fast and frugal metric.
Therefore, A and B cannot be “incompatible” questions: The presence of A does not act as a context for B. There is no potentiality
that becomes concrete once the other piece is present.
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
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Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
It is simply that each metric has its own memories or heuristics.
P&B dismiss such an explanation, but it can be related to Herbert
Simon’s “bounded rationality.” For Simon, the person has limited cognitive resources, and hence has to use rules of thumb to make
decisions. Such rules of thumb are fast ways to look at a problem
when the stakes are not high. That is, such rules of thumb are actually
efficient, which explains why standard economists welcome Simon’s
notion of bounded rationality. To wit, if the stakes are high, people
usually review and replace the existing rule of thumb (Baron 2008,
Ch. 6). People are even ready to correct their errors of logical reasoning, as in the Wason selection task or the famous Linda in the conjunction fallacy, once they are given extra time for reflection and cognition.
Such correction would not be possible if the order effect were
similar to the context effect in a substantial (homologous) sense.
But what is “context”? It is the outcome of the neural principle
when it specifies the background, which allows the comprehension
of the foreground data; similar to how the brain selects the background in Rubin’s vase. The principle that specifies the background
cannot be “corrected” by data because it is not a datum. The background acts as a context that affords “meaning” or, as discussed
earlier, a “potentiality” of what the observed thing can become.
An example of the context effect is the “gain frame” versus the
“loss frame” in the Asian disease experiment: Two medical remedies are presented to deal with an Asian disease: one remedy has a
certain outcome in terms of saved lives or prevention from death,
whereas the other has uncertain outcomes. The uncertain remedy
has an expected value equal to the certain remedy. Subjects
usually chose the certain remedy if the results were couched as
“saving lives” rather than “preventing death,” that is, using the
“gain frame,” but subjects usually chose the uncertain remedy if
the results were couched in the other way, that is, using the
“loss frame.”
LeBoeuf and Shafir (2003) showed that subjects, once exposed to
a context, would not generally “correct” the choice if given more
time to reflect, even if such subjects had scored high on the “need
for cognition” test, that is, enjoyed cognitive pursuits (Cacioppo &
Petty 1982). This confirms that context is not a datum to be corrected. The context is rather a background that provides meaning
to the foreground, content. The context and content are asymmetrical, that is, stand dependently on each other. This is contrary to the
Gore/Clinton test in which A and B are symmetrical, that is, stand
independently of each other. Therefore, the context effect is radically different from the order effect.
Therefore, P&B commit an identificational slip: they confuse
analogies with homologies. The same technique (whether CP or
QP) cannot apply to the context effect, on the one hand, and to
the order effect and its associated biases, on the other.
The cognitive economy: The probabilistic turn
in psychology and human cognition
doi:10.1017/S0140525X12003019
Petko Kuseva,b and Paul van Schaikc
a
Department of Psychology, Kingston University London, London KT1 2EE,
United Kingdom; bCity University, London EC1V OHB, United Kingdom;
c
Department of Psychology, Teesside University, Middlesbrough TS1 3BA,
United Kingdom.
[email protected]
[email protected]
http://kusev.co.uk
http://sss-studnet.tees.ac.uk/psychology/staff/Paul_vs/index.htm
Abstract: According to the foundations of economic theory, agents have
stable and coherent “global” preferences that guide their choices among
alternatives. However, people are constrained by information-processing
and memory limitations and hence have a propensity to avoid cognitive load.
We propose that this in turn will encourage them to respond to “local”
preferences and goals influenced by context and memory representations.
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One of the most significant current discussions in economics and
psychology is about the (lack of) link between normative (what we
should do, based on probability and logic) and descriptive theories
(what we do) of decision making. An obvious challenge for and
advantage of good theory in decision making is that it is general,
being able to account for both normative (rational) and descriptive
psychological mechanisms and assumptions (Kusev et al. 2009).
According to the foundation of economic theory, people have
stable and coherent “global” preferences that guide their choices
among alternatives varying in risk and reward. In all their variations
and formulations, normative utility theory (von Neumann & Morgenstern 1947), and descriptive prospect theory (Kahneman &
Tversky 1979; Tversky & Kahneman 1992) share this assumption
(Kusev et al. 2009). However, a consistent claim from behavioral
decision researchers is that, contrary to the assumptions of classical
economics, preferences are not stable and inherent in individuals
but are “locally” constructed “on the fly” and are strongly influenced
by context and the available choice options (e.g., Kusev et al. 2009;
2012a; 2012b; Slovic 1995). For example, the preference reversal
phenomenon (Lichtenstein & Slovic 1971; 1973) suggests that no
stable pattern of preference underlies even basic choices; in other
words, consistent trade-offs between lotteries with different probabilities and values are not made. Accordingly, we shall present
some evidence for violations of the classical probability framework,
inspired by existing research in judgment and decision making,
and also review plausible sources for understanding human decision
making: specifically simplicity in human information processing,
based on local decision making goals and strategies.
Theorists in cognitive science achieve general theoretical propositions, based on the following assumptions: (1) the cognitive
systems solve problems, optimally, given environmental and processing constrains (Anderson 1990; 1991); therefore, the objective is to
understand the structure of the problem from the point of view of
cognitive systems and (2) cognitive goals determine choice behavior: when a general cognitive goal is intractable, a more specific cognitive goal, relevant to achieving the general goal, may be tractable
(Oaksford & Chater 2007; 2009). For example, all local goals are
assumed to be relevant to more general goals, such as maximizing
expected utility. The observation that the local goals may be optimized as surrogates for the larger aims of the cognitive system
raises another important question about the use of rational
models of human cognition. Specifically, Oaksford and Chater
(2007; 2009) propose that optimality is not the same as rationality.
The fact that a model involves optimization does not necessarily
imply a rational model; rationality requires that local goals are (1)
relevant to general goals and (2) reasonable. Here we make a
very simple assumption about how and whether the cognitive
system optimizes. We assume that the cognitive system simplifies
and adopts “local” goals, and that these goals will be influenced
by contextual and memory representations. Accordingly, we
argue, strengthening such an account could provide a challenge
to classical probability approach.
Whereas some phenomena in judgment and decision making
systematically violate basic probabilities rules – classic examples
include the conjunction fallacy (Tversky & Kahneman 1983),
the disjunction effect (Tversky & Shafir 1992), the subadditivity
principle (e.g., Tversky & Koehler 1994), and the preference
reversal phenomenon (Lichtenstein & Slovic 1971; 1973; Slovic
1995) – the simplicity framework suggested in this commentary
argues that people are constrained by information-processing
and memory limitations, and hence have a propensity to avoid
cognitive load. Research in judgment and decision making
demonstrates that by focusing on local goals (e.g., the representativeness heuristic) people may violate principles of classical probability theory (e.g., fallacies in which specific conditions are
assumed to be more probable than a single general one). For
example, the independence assumption states that the occurrence
of one event makes it neither more nor less probable that the
other occurs; examples of violation include the conjunction
fallacy (Tversky & Kahneman 1983), the disjunction effect
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
(Tversky & Shafir 1992), and the familiarity bias (e.g., Fox &
Levav 2000; Tversky & Koehler 1994). One plausible account
for these effects, as an alternative to classical logic, classical probability, and the classical information-processing paradigm, is
quantum probability theory (Busemeyer & Wang 2007; Busemeyer et al. 2006; 2011; target article, sects. 1 and 2). These
authors argue that the “classical” view forces highly restrictive
assumptions on the representation of the complex cognitive
system. In particular, they suggest that (1) the brain is a
complex information-processing system with a large number of
unobservable states, (2) the brain is highly sensitive to context
and, finally, (3) the measurements that we obtain from the brain
are noisy and subject to uncertainty. Pothos & Busemeyer
(P&B) show that quantum probability theory allows the modeling
of decision-making phenomena (e.g., the conjunction fallacy and
violations of the sure-thing principle), going beyond classical probability theory. This is because quantum probability theory can
account for context- and order-dependence of human behavior.
We conclude that the cognitive system is likely to respond to
“local” goals (that might be tractable) influenced by memory representations and context that may be indicative of probability and
frequency judgments. Therefore, in contrast to classical probability theory, quantum probability theory has the potential to
account for context- and order-dependent behavior that is indicative of human propensity to adopt local goals. Moreover, there is
mounting evidence for this type of behavior and simple mechanisms ruling human behavior (Kusev et al. 2011). Therefore,
quantum probability has the potential to account for cognitive
economy in many domains of human cognition.
Quantum models of cognition as Orwellian
newspeak
doi:10.1017/S0140525X12003020
Michael D. Leea and Wolf Vanpaemelb
a
Department of Cognitive Sciences, University of California Irvine, Irvine CA
92697-5100; bFaculty of Psychology and Educational Sciences, University of
Leuven, 3000 Leuven, Belgium.
[email protected]
[email protected]
http://faculty.sites.uci.edu/mdlee/publications/
http://ppw.kuleuven.be/okp/people/wolf_vanpaemel/
Abstract: Faced with probabilistic relationships between causes and
effects, quantum theory assumes that deterministic causes do not exist,
and that only incomplete probabilistic expressions of knowledge are
possible. As in its application to physics, this fundamental epistemological
stance severely limits the ability of quantum theory to provide insight and
understanding in human cognition.
Some physicists have serious objections to quantum theory. Their
dissent does not relate to ontology – there is broad acceptance that
quantum theory has led to surprising and accurate predictions
about the physical world, with useful applied results – but,
rather, to epistemology. As Pothos & Busemeyer (P&B) make
clear in their target article, they are not making ontological
claims about the brain being quantum but are arguing for the epistemological promise of the quantum approach offering psychological understanding of “the fundamental why and how
questions of cognitive process” (sect. 1.2). Therefore, the criticisms from physics are directly relevant.
The physicists’ argument against the quantum approach to modeling is clearly explained by Jaynes (2003, pp. 327–28). If there is an
effect E that does not occur unless some condition C is present, it
seems natural to infer that the condition is a necessary causative
agent for the effect. If, however, the condition does not always
lead to the effect, it seems natural to infer that there must be
some other causative factor F, which has not yet been understood.
This is the approach taken by classical probability theory. Asking the
Gore question C causes an opinion E to be produced. If the opinion
is different when the Clinton question F is asked first, then the conditional probability p(E|C) must be extended to p(E|C,F) to take
account of the relevant additional information in factor or context
F. Quantum theory, as advocated by the target article, takes a fundamentally different approach. Faced with probabilistic relationships between causes and effects, quantum theory assumes that
deterministic causes do not exist, and that only incomplete probabilistic expressions of knowledge are possible. Rather than trying
to understand the exact nature of the influence of the relevant
factor F, quantum theory simply assumes this level of understanding is beyond its scope.
We think that it is reasonable to have reservations when dealing
with something as complicated as human cognition, about
whether it is practically feasible to isolate all the relevant factors
and fully understand their interactions, but that is not the issue
here. In sharp contrast to classical probability, adopting the
quantum approach makes it nonsensical even to aim for this
understanding. As Jaynes (2003, p. 328) puts it: “The mathematical formalism of present quantum theory, like Orwellian newspeak, does not even provide a vocabulary with which one could
ask such a question.” It is the difference between, in practice,
not having the understanding of physics needed to control deterministically the outcome of a coin flip, versus assuming, in principle, that such a level of understanding can never be reached
(i.e., that even with sophisticated instruments for applying
forces and measuring initial conditions, the outcome of a physical
coin flip is inherently probabilistic). The deterministic effects of
order, context, and the other factors highlighted in the target
article on human cognition might be hard to understand, but
adopting the quantum approach requires us to stop trying.
The target article seems to acknowledge this hamstringing, when
it argues that the basic quantum property of superposition is “an
intuitive way to characterize the fuzziness (the conflict, ambiguity,
and ambivalence) of everyday thought.” (sect. 1.1) But should characterizing indefiniteness be the objective when building models of
cognition? Surely the goal of modeling cognition is to sharpen our
understanding, and remove the fuzziness, rather than to replicate it
in the modeling machinery. The fact that people cannot unpack the
relevant contexts or other factors that mediate their cognitive processes does not mean that our models must be similarly unable to
do so. The same basic issue plays out in discussion of entanglement,
which makes it impossible to construct a complete joint distribution
between two variables, and therefore impossible to model how they
interact with each other. Instead, the quantum approach implants
indefiniteness about how cognition works directly into the modeling
framework, and there is neither the possibility nor need to work
hard to build better models that remove the confusion.
The target article argues that the acid test is “whether there are
situations where the distinctive features of QP theory provide a
more accurate and elegant explanation for empirical data.” (sect.
1.2) Viewing quantum theory as newspeak implies that it cannot,
in principle, lead to complete explanations. We think the worked
examples in the target article bear this out. There are striking disconnects between the descriptions of the formal workings of the
models, and their psychological interpretations. The former seem
devoid of psychological content and the latter seem vague and
verbal. For example, the projection of lines onto subspaces that
models the conjunction fallacy, with its mathematical precision, is
“explained” as “a kind of abstraction process, so that the projection
on to the feminist subspace loses some of the details about Linda.”
(sect. 3.1)
Sloman’s blurb for the Busemeyer and Bruza (2012) book on
quantum models of cognition and decision says “Mathematical
models of cognition so often seem like mere formal exercises.
Quantum theory is a rare exception. Without sacrificing formal
rigor, it captures deep insights about the workings of the mind
with elegant simplicity.” We think that is exactly the wrong way
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Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
around. Quantum models of cognition offer formal exercises that
might produce impressive fits to data but, by their founding
assumptions, cannot offer some of the most basic insights into
the causes, effects, and relevant factors that underlie the workings
of human cognition.
Jaynes (1993, p. 269) puts the physicists’ epistemological dissent
bluntly, saying “I am convinced, as were Einstein and Schrödinger, that the major obstacle that has prevented any real progress
in our understanding of Nature since 1927, is the Copenhagen
Interpretation of Quantum Theory. This theory is now 65 years
old, it has long since ceased to be productive, and it is time for
its retirement.” It would be unfortunate if a theory ready for retirement in its professional field of physics were to enjoy a second hobbyist career in psychology.
Grounding quantum probability in
psychological mechanism
doi:10.1017/S0140525X12003147
Bradley C. Love
University College London, Cognitive, Perceptual and Brain Sciences, London
WC1H 0AP, United Kingdom.
[email protected]
bradlove.org
Abstract: Pothos & Busemeyer (P&B) provide a compelling case that
quantum probability (QP) theory is a better match to human judgment
than is classical probability (CP) theory. However, any theory (QP, CP,
or other) phrased solely at the computational level runs the risk of being
underconstrained. One suggestion is to ground QP accounts in
mechanism, to leverage a wide range of process-level data.
Pothos & Busemeyer (P&B) make clear that quantum probability
(QP) theory offers a rich array of theoretical constructs, such as
superposition, entanglement, incompatibility, and interference,
which can help explain human judgment. The authors illustrate
how these concepts, which are strongly contrasted with the
basic tenets of classical probability (CP) theory, can be used to
accommodate aspects of human choice that deviate from normative CP accounts. For example, the conjunction fallacy is
explained in terms of incompatible questions requiring sequential
evaluation, which induces an interference effect.
Although new frameworks can provide novel insights, one worry
is that QP will recapitulate some of the shortcomings of rational CP
approaches by sticking to a computational-level analysis. To the
authors’ credit, they acknowledge how notions of optimality in
CP approaches can be impoverished and not match the goals of
the decision maker. However, these criticisms largely serve to question CP’s status as the preferred normative account rather than
question the wisdom of eschewing process-level considerations in
favor of a computational-level analysis.
In a recent article with Jones (Jones & Love 2011), we, too, critiqued rational (Bayesian) CP approaches to explaining human
cognition, but our critique was broader in scope. Although
many of our points are particular to the rational Bayesian
program (which we refer to as “Bayesian Fundamentalism”),
some of the central critiques apply equally well to any approach
largely formulated at the computational level. The basic issue is
that such accounts wall off a tremendous amount of related data
and theory in the cognitive sciences, including work in attention,
executive control, embodiment, and cognitive neuroscience, as
well as any study using response time measures. It seems unlikely
that a complete theory of cognition or decision making can be formulated when neglecting these insights and important constraints.
The suggestion offered in Jones and Love (2011), which we
referred to as “Bayesian Enlightenment,” is to integrate probability and mechanistic approaches. In the context of QP, one
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could imagine construing operations, such as projections to subspaces, as psychological operations that unfold in time, may
have brain correlates, be limited in capacity, and change over
development. Such an approach would retain the distinctive
characteristics of QP while linking to existing theory and data.
Grounding QP in mechanism may offer a number of other
advantages, such as better motivating the assumptions (that are
psychological in nature) that make QP successful. Many of the
effects considered in the target article require assumptions on
the order in which statements are considered and the role
context plays. These topics may be addressed in a principled
manner when situated within a mechanism that aims to explain
shifts in focus or attention. Such mechanistic models would also
make clear what role QP plays in accounting for the results, as
opposed to the ancillary assumptions.
The authors note that one key challenge is to anticipate new findings rather than simply accommodate existing data. Grounding QP
ideas in mechanism may facilitate making a priori predictions. Once
the move to mechanism is made, second generation questions can
be asked, such as which QP model best accounts for human judgment. My guess is that moving away from evaluating general frameworks to testing specific proposals will hasten progress. As the
authors note, it is very difficult to invalidate an entire framework,
as ancillary assumptions can always be made (e.g., CP models can
be modified to account for the main findings in the target
article). In contrast, particular models can be evaluated using
model selection procedures.
My prediction is that moving toward evaluating particular
models grounded in mechanism will lead to a rapprochement
between QP and CP approaches. For a view that allows for superposition, many aspects of the QP are very rigid. For example,
according to the approach advocated by the authors, statements
are either compatible or incompatible. One possibility is that successful models will be more fluid and include a mixture of states,
which is a notion from CP. Given the complexities of human cognition and decision making, it would be surprising if one unadulterated formalism carried the day. Although physics
undergraduates may complain about how confusing QP is,
human cognition will likely prove more vexing.
Cognition in Hilbert space
doi:10.1017/S0140525X1200283X
Bruce James MacLennan
Department of Electrical Engineering and Computer Science, University of
Tennessee, Knoxville, Knoxville, TN 37996.
[email protected]
http://web.eecs.utk.edu/∼mclennan/
Abstract: Use of quantum probability as a top-down model of cognition
will be enhanced by consideration of the underlying complex-valued
wave function, which allows a better account of interference effects and
of the structure of learned and ad hoc question operators. Furthermore,
the treatment of incompatible questions can be made more quantitative
by analyzing them as non-commutative operators.
Pothos & Busemeyer (P&B) argue for the application of quantum
probability (QP) theory to cognitive modeling in a function-first or
top-down approach that begins with the postulation of vectors in a
low-dimensional space (sect. 2.1), but consideration of the highdimensional complex-valued wave function underlying the state
vector will expand the value of QP in cognitive science. To this
end, we should import two premises from quantum mechanics.
The first is that the fundamental reality is the wave function. In
cognitive science, this corresponds to postulating spatially distributed patterns of neural activity as the elements of the cognitive
state space. Therefore, the basis vectors used in QP are basis functions for an infinite (or very high) dimensional Hilbert space. The
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
second premise is that the wave function is complex valued and
that wave functions combine with complex coefficients, which is
the main reason for interference and other non-classical phenomena. The authors acknowledge this (sects. 2.3, 3.3, Appendix), but
they do not make explicit use of complex numbers in the target
article.
There are several possible analogs in neurophysiology of the
complex-valued wave function, but perhaps the most obvious is
the distribution of neural activity across a region of cortex; even
a square millimeter of which can have hundreds of thousands of
neurons. The dynamics are defined by a time-varying Hamiltonian, with each eigenstate being a spatial distribution of
neurons firing at a particular rate. The most direct representations
of the magnitude and phase (or argument) of a complex quantity
are the rate and relative phase of neural impulses.
The target article specifies that a decision corresponds to
measurement of a quantum state, which projects the cognitive
state into a corresponding eigenspace, but it is informative to consider possible mechanisms. For example, the need to act definitely
(such as coming to a conclusion to answer a question) can lead to
mutually competitive mechanisms, such as among the minicolumns
in a macrocolumn, which create dynamic attractors corresponding
to measurement eigenspaces. Approach to the attractor amplifies
certain patterns of activity at the expense of others. Orthogonal projectors filter the neural activity and win the competition with a
probability proportional to the squared amplitude of their inner
products with the wave function. (In the case in which impulse
phases encode complex phases, matching occurs when the phases
are delayed in such a way that the impulses reinforce.) The
winner may positively reinforce its matched signal components
while the loser negatively reinforces its matched components.
Regardless of mechanism, during collapse, the energy of the
observed eigenstate of the question (measurement) operator captures the energy of the orthogonal eigenstates (this is the effect
of renormalization). The projection switches a jumble of frequencies and phases into a smaller, more coherent collection, corresponding to the outcome (observed) eigenspace. This
competition also explains the prioritization of more likely outcomes
(sect. 3.1).
The target article (sect. 2.1) suggests that a QP model of cognition begins by postulating basis vectors and qualitative angles
between alternative question bases (significantly, only real
rotations are discussed). As a consequence, a QP model is
treated as a low-dimensional vector space. This is a reasonable,
top-down strategy for defining a QP cognitive model, but it can
be misleading. There is no reason to suppose that particular question bases are inherent in a cognitive Hilbert space. There may be
a small number of “hard-wired” questions, such as fight-or-flight,
but the vast majority is learned. Certainly this is the case for questions corresponding to lexical categories such as (un-)happy and
(un-)employed.
Investigation of the dynamics of cognitive wave function collapse would illuminate the mechanisms of decision making, but
also the processes by which observables are organized. This
would allow modeling of changes in the question bases, either
temporary through context effects, or longer lasting through
learning. Furthermore, many question bases are ad hoc, as
when we ask, “Do you admire Telemachus in the Odyssey?”
How such ad hoc projectors are organized requires looking
beneath a priori basis vectors to the underlying neural wave functions and the processes shaping them.
Certainly one of the most interesting consequences of applying
to QP to cognition is the analysis of incompatible questions. The
approach described in the target article (sect. 2.2) begins by postulating that incompatible questions correspond to alternative
bases for a vector space. The qualitative angle between the question bases is estimated by a priori analysis of whether the questions
interfere with each other.
In quantum mechanics, however, the uncertainty principle is a
consequence of non-commuting measurement operators, and the
degree of non-commutativity can be quantified. Two measurement operators P and Q commute if PQ = QP, that is, if the operator PQ−QP is identically 0. If they fail to commute, then PQ−QP
measures the degree of non-commutativity, which is expressed in
quantum mechanics by the commutator [P,Q] = PQ−QP. It is
relatively easy
that this implies an uncertainty relation:
to show
DPDQ ≥ k P, Q l. That is, the product of the uncertainties on
a state is bounded below by the absolute mean value of the commutator
on the state. Suppose
H is a measurement that returns 1
for happy l and
0 for unhappy l, and
E is a measurement that
returns 1 for employedl and 0 for unemployedl. If
employedl = ahappy l + bunhappy l,
0 1
then the commutator is [H, E] = ab
−1 0
and the magnitude of the commutator applied to an arbitrary state |ψ〉 is ||[H,E]| ψ〉||=|ab|.
Might we design experiments to measure the commutators and
so quantify incompatibility among questions? Certainly there are
difficulties, such as making independent measurements of both
PQ and QP for a single subject, or accounting for intersubject
variability in question operators. But making such measurements
would put more quantitative teeth into QP as a cognitive model.
Processes models, environmental analyses,
and cognitive architectures: Quo vadis
quantum probability theory?
doi:10.1017/S0140525X12003032
Julian N. Marewski and Ulrich Hoffrage
University of Lausanne, Quartier UNIL-Dorigny, 1015 Lausanne, Switzerland.
[email protected]
[email protected]
http://www.hec.unil.ch/people/jmarewski
http://www.hec.unil.ch/people/uhoffrage
Abstract: A lot of research in cognition and decision making suffers from a
lack of formalism. The quantum probability program could help to
improve this situation, but we wonder whether it would provide even
more added value if its presumed focus on outcome models were
complemented by process models that are, ideally, informed by
ecological analyses and integrated into cognitive architectures.
In the cognitive and decision sciences, much research suffers from
a lack of formalism. This is particularly the case for qualitative
accounts of behavior proposed, for instance, within the heuristics-and-biases framework (Kahneman et al. 1982), or within
related dual process theories of cognition (Sloman 1996). We
applaud Pothos & Busemeyer’s (P&B’s) attempt to promote a
formal framework that contributes to remedying this shortcoming
and that has a high potential for being innovative and useful. With
that being said, we take issue with three aspects of the quantum
probability (QP) program.
First, we posit that outcome models should be complemented by
process models. What level of description do P&B envision for QP
models? One of the central goals of many psychological theories is to
describe cognitive processes. In contrast, behavioral economists and
cognitive scientists working with, for example, Bayesian models
(e.g., Griffiths et al. 2008) focus on predicting the outcomes of
behavior, without necessarily aspiring to provide plausible accounts
of the underlying processes (Berg & Gigerenzer 2010). We worry
that the QP program falls into this class of outcome-oriented (or
as-if) models, banishing algorithmic-level accounts of memory,
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Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
perceptual, motivational, and decisional processes into the behaviorist’s black box. We challenge P&B to demonstrate how the QP framework can contribute to developing process models such as those
proposed by the adaptive toolbox program (Gigerenzer & Selten
2001). The repertoire of fast-and-frugal heuristics developed in
this program includes algorithmic rules that specify how the cognitive system searches for information, when it will stop searching, and
how it will combine the acquired pieces of information in order to
make a decision. Consider, for instance, the priority heuristic
(Brandstätter et al. 2006). This simple lexicographic strategy for
making risky choices between gambles is composed of three rules
that operate on the probability and outcome vectors constituting
the gambles. For example, the first rule prescribes in what order
the information contained in these vectors is searched. The rules
of these and other heuristics have been used to motivate process
predictions about reaction times (Bröder & Gaissmaier 2007), eye
movements (Reisen et al. 2008), functional magnetic resonance
imaging (fMRI) data (Volz et al. 2010), and the amount of information looked up on a computer screen (Johnson et al. 2008).
Process models help to gain a deeper understanding of behavior,
and their predictions – albeit not always free of scholarly disputes
(e.g., Marewski et al. 2010; 2011) – can eventually differentiate
among competing theories that make identical outcome predictions.
Second, we posit that the process models we ask for should be
informed by an analysis of the structure of the environment – after
all, it takes two to tango. As has been stressed by many cognitive
modelers (Anderson & Schooler 1991; Dougherty et al. 1999;
Gigerenzer et al. 1991; Oaksford & Chater 1998; Simon 1956),
the human cognitive system is adapted to its ecology. To use
Brunswik’s (1964) metaphor, mind and environment are like
spouses who have co-evolved, who mutually shaped each other,
and who cannot and should not be separated. The heuristicsand-biases program does not consider the situated nature of cognitive processes; rather, typically its heuristics are hypothesized to
be invoked independent of context and there is no ecological
analysis specifying how performance depends on environmental
factors (Gigerenzer 1996). P&B use the heuristics that have
been proposed in the heuristics-and-bias tradition to motivate
the QP program, pointing out that these heuristics reveal the
insufficiencies of the classical probability program. While the
QP program addresses some of these insufficiencies, others
might be remedied by more explicitly considering (1) how
environmental structures have shaped cognitive processes, (2)
how such structures can select cognitive processes, and (3) how
these processes, once selected, perform differentially in environments with different statistical structures. For instance, the shape
of the distribution of environmental variables influences the performance of the QuickEst and social-circle heuristics, two fastand-frugal heuristics for estimating quantities and frequencies,
respectively (Hertwig et al. 1999; Pachur et al. 2013). More
examples of how the functioning of cognitive processes depends
on environmental structures and how an analysis of the environment can assist researchers to develop – what one might term –
ecological process models can be found in Ecological Rationality:
Intelligence in the World (Todd et al. 2012) and in Simple Heuristics in a Social World (Hertwig et al. 2013).
Third, we posit that such ecological process models should be
integrated into cognitive architectures. As P&B correctly point
out, any theory requires constraints. One way to constrain psychological theories consists of introducing axioms and positing that
cognition adheres to these axioms. Another way to constrain theories requires formulating psychologically grounded guiding principles that reflect empirical observations about behavior (cf.
Anderson & Lebiere 2003). Consistent with the latter approach,
we and others (Marewski & Mehlhorn 2011; Marewski &
Schooler 2011; Nellen 2003; Schooler & Hertwig 2005) have
modeled a few of the fast-and-frugal heuristics – including their
dependencies on environmental, mnemonic, and perceptual variables – with the ACT-R architecture (Anderson et al. 2004). This
architecture integrates models of memory, perception, and other
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components of cognition into one unified theory. In doing so,
ACT-R incorporates the findings of decades of empirical work
into computer code and mathematical equations, which, in turn,
helps to constrain new models of cognition – including heuristics –
when these are implemented in the architecture. We wonder to
what extent the QP program’s axiomatic approach provides
added value for constraining ecological process models beyond
what can be achieved by implementing them into psychologically
plausible architectures. We hasten to add that we also wonder
whether the QP program’s axiomatic approach might aid, conversely, in further refining such architectures.
To conclude, we believe the QP program may do a lot of good.
However, we ask ourselves whether it could do even better if (1)
its presumed focus on outcome models were complemented by
process models that (2) are, ideally, informed by ecological analyses and (3) integrated into cognitive architectures.
The implicit possibility of dualism in quantum
probabilistic cognitive modeling
doi:10.1017/S0140525X12003044
Donald Mender
Department of Psychiatry, Yale University, New Haven, CT 06511.
[email protected]
Abstract: Pothos & Busemeyer (P&B) argue convincingly that quantum
probability offers an improvement over classical Bayesian probability in
modeling the empirical data of cognitive science. However, a weakness
related to restrictions on the dimensionality of incompatible physical
observables flows from the authors’ “agnosticism” regarding quantum
processes in neural substrates underlying cognition. Addressing this
problem will require either future research findings validating quantum
neurophysics or theoretical expansion of the uncertainty principle as a
new, neurocognitively contextualized, “local” symmetry.
Pothos & Busemeyer’s (P&B’s) answer to the question posed by
the title of their article, “Can quantum probability provide a
new direction for cognitive modeling?” addresses almost all the
relevant issues. In particular, from an empirical viewpoint, the
authors offer substantial evidence that quantum probability, explicated through its most transparent mathematical representation in
generalized phase space, fits a great mass of cognitive data more
closely than does classical Bayesian probability. P&B show
through numerous examples that this superior fit applies with
respect to quantum probability’s superpositional characteristics,
unitarily time-evolved interfererence effects, entangled composition, and mutual incompatibility of precision in measurements
entailing ordered context for certain pairs of observables enlisted
as operators.
However, the word “certain” in the last sentence points toward
one residual hole in the authors’ agenda. That defect is their professed “agnosticism” regarding possibilities for physical embodiment
of quantum probability in the thinking brain. Such agnosticism
opens up an anarchical lack of limitation in choosing those particular
pairs of observables that must be treated as mutually incompatible
in cognitive applications of quantum probability.
Physicists as a rule take incompatibility between operator pairs
to be a concept whose applicability is restricted only to “certain” –
that is, canonically conjugate physical – coordinates. Examples of
canonically conjugate coordinate pairs include energy yoked to
time and momentum yoked to distance. Each such pair yields a
quantitative product with the qualitative dimensionality of action
applicable to Planck’s constant at the foundation of quantum
mechanics and to the path-integrated functionals of quantum
field theory.
If P&B were to follow the restrictive practices of physicists by
limiting their choices of operator pairs to canonically conjugate
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
observables, then physical embodiment of mental cognition in
corporeal brain processes would be easy to infer. However, the
authors also propose to consider for quantum probability theory
qualitatively mentalistic observables, for example “happiness” or
“feminism,” entirely abstracted from the material brain, and
non-corporeally applied to mind. Some such observables might
conceivably be posited as mutually incompatible, non-commutatively order-sensitive sets of operators, even if they have been
arbitrarily untethered from the physical anchor of canonical conjugation and the dimensionality of action, but that kind of rash
detachment from the canonically conjugate constraints of
physics could condemn cognitive science to a new form of Cartesian dualism. There would issue mentalistic forms of superposable
probability amplitudes pertaining to properties and extended in
“spaces” whose unitary “evolutions,” mutual interference “effects,”
entangled “non-locality,” and Fourier-dual “measurement” uncertainties lack any necessary connections to established physical observables and space times (or, alternatively, to their spin-networked
foundations per loop quantum gravity) in which brains exist.
These caveats are especially germane if one stipulates that cognition, beyond the empirical question of its conformity to
quantum probability, is formally quantum computational. The
latter possibility remains an open biophysical question, insofar
as classically engineered hardware can only imperfectly simulate
but not fully support a quantum mode of abstract computation.
As P&B mention at the end of their article, nobody really
knows yet whether the brain’s biophysics are compatible with
the coherence of putative quantum wetware processes. Some,
such as Tegmark (2000), have championed principled thermodynamic objections to any such brain capabilities. Rebuttals by
Hameroff and others (Hagan et al. 2002) to Tegmark’s concerns
about thermal decoherence have centered on two classes of argument. First, recent experimental evidence that quantum physics
plays a role in photosynthesis (Engel et al. 2007) has been
invoked against the idea that temperatures far colder than
those found in living biological systems are needed to sustain
quantum-coherent states. Second, proposals for thermal insulators that might protect physical quantum processes in the brain
have been advanced. These include, for example, ordered
water and pumped phonons, but to date none have been
proven experimentally to operate in the relevant biological
contexts.
Let us suppose that Tegmark’s opponents turn out to be
wrong, and quantum wetware substrates are nowhere to be
found in neural tissue. If it transpires that the brain cannot
sustain a quantum-biophysical “wetware” for cognition, then it
will be hard to see how quantum-probabilistic “software” can
offer a compelling perspective on cognitive phenomena within
the framework of known metaphysical monisms. Thereafter
those cognitive phenomena whose statistics conform more
closely to abstract quantum probability than to classical Bayesian
probability will be stuck between a rock and a hard place. Either
quantum probability’s disembodied elegance will have to be
given up, or a new kind of dualism will be needed for cognitive
science.
Making unified sense of that dualism will perhaps require some
kind of trans-physical, post-Cartesian gauge, mimicking the generators of standard physical forces through coordination among
“certain” local symmetries, but bridging quantitative intervals in
“spacetimes” that, unlike the standard gauge fields of today’s
physics, are in a qualitative sense not purely physical. The new
“gauge” would be required both to contact and to transcend corporeal physics while either obeying accepted physical laws or compensating against its own violations of those laws. A crucial
underlying supposition in constructing such a psychophysical
gauge might be the “local” character of canonically conjugate
Fourier-duality as a kind of symmetry embedded within the
“metrical” contexts of qualitatively neurocognitive “spaces” and
“times.” The specifics of that psychophysical gauge’s design,
which will have to confront head on Chalmers’ “hard problem”
of generally relating perceptual qualia and physical quantitation
(Chalmers 1995), could well prove to be the most daunting neuroscientific challenge for any quantum probabilistic cognitive
paradigm.
What are the mechanics of quantum
cognition?
doi:10.1017/S0140525X12003056
Daniel Joseph Navarroa and Ian Fussb
a
School of Psychology, University of Adelaide, SA 5005 Australia; bSchool of
Electrical Engineering, University of Adelaide, SA 5005 Australia.
[email protected]
[email protected]
Abstract: Pothos & Busemeyer (P&B) argue that quantum probability
(QP) provides a descriptive model of behavior and can also provide a
rational analysis of a task. We discuss QP models using Marr’s levels of
analysis, arguing that they make most sense as algorithmic level theories.
We also highlight the importance of having clear interpretations for
basic mechanisms such as interference.
What kind of explanation does a cognitive model offer? A standard
way of approaching this question is to use Marr’s (1982) three
levels of explanation. A “computational analysis” provides an
abstract description of the problem that the learner must solve,
along with a normative account of how that problem should be
solved. Bayesian models of cognition are usually computational
level explanations. An “algorithmic level” explanation describes a
mechanistic process that would produce human-like behavior in
some task. Most traditional information-processing models and
many connectionist models lie at this level of explanation.
Finally, “implementation level” explanations propose a low-level
physical explanation of how the brain might perform the computations that are required. These are the kinds of models typically
pursued in cognitive neuroscience.
Whereabouts in this classification scheme should we place the
quantum probability (QP) framework? The implementation level
is the simplest to consider. Pothos & Busemeyer (P&B) explicitly
disavow any implementation level interpretation of these
models; making a clear distinction between their work on the
formal modeling of cognition using a quantum formalism and
those researchers (e.g., Hameroff 1998) who argue that neural
function should be modeled as a quantum physical system. We
agree with this distinction.
Should QP models be treated as computational level analyses?
Although P&B make explicit comparisons to classical probability
and to Bayesian models, we do not think it makes sense to treat
QP models as computational level analyses. The critical characteristic of a computational analysis is to specify what problem the
learner is solving, and to present a normative account of how
that problem should be solved. Bayesian models work well as
computational analyses because of the fact that classical probability provides good rules for probabilistic inference in everyday
life. In discussing this issue, P&B point to problems associated
with statistical decision theory (e.g., that Dutch books are possible
in some cases), or to well- known issues with the Kolmogorov
axioms (e.g., sample spaces are hard to define in real world contexts). However, in our view their discussion misses the forest for
the trees: showing that classical probability has limitations does
not establish QP as a plausible alternative. There is a good reason
why statistics is built on top of classical probability and not
quantum probability: it is the right tool for the job of defining normative inferences in everyday data analysis. In contrast, although
there are such things as “quantum t-tests” (e.g., Kumagai &
Hayashi 2011), they have yet to find a natural role within everyday
statistical analysis. It is possible that such usage may emerge in time,
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
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Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
but we think this is unlikely, simply because the situations to which
such tools are applicable (e.g., data follow a quantum Gaussian distribution) do not arise very often when one is analyzing real data.
Until statistical tools based on QP find a place in everyday data
analysis, we remain unconvinced that QP makes sense as a normative account of everyday inference.
Regarding the algorithmic level, we think that P&B are on more
solid ground: there is some justification for thinking about QP
models as mechanistic accounts. Consider the model used to
account for Shafir and Tversky’s (1992) data on the prisoner’s
dilemma. It relies on an interference effect to account for the
fact that participants defect whenever the opponent’s action is
known but cooperate when it is unknown. This interference
does not emerge as part of an optimal solution to the inference
problem given to the decision maker, nor is it characterized at a
neural level. It is clearly intended to refer to a psychological mechanism of some kind.
In view of this, a mechanistic view of QP seems to provide the
right way forward, but at times it is difficult to understand what
the mechanisms actually are. To take a simple example, why are
some questions incompatible and others are compatible? P&B
suggest that “[a] heuristic guide of whether some questions should
be considered compatible or not is whether clarifying one is
expected to interfere with the evaluation of the other” (sect. 2.2).
This seems sensible, but it begs the question. One is naturally led
to ask why some psychological states interfere and others do not.
This is difficult to answer because the QP formalism is silent on
how its central constructs (e.g., interference) map onto psychological
mechanisms. In our own work (Fuss & Navarro, in press) we have
explored this issue in regards to the dynamic equations that describe
how quantum states change over time. Specifically, we have sought
to describe how these equations could arise from mechanistic processes, but our solution is specific to a particular class of models
and we do not claim to have solved the problem in general. In
our view, understanding how formalisms map onto mechanisms is
one of the biggest open questions within the QP framework.
In short, we think that the potential in QP lies in developing
sensible, interpretable psychological mechanisms that can
account for the otherwise puzzling inconsistencies in human
decision making. It might be that human cognition cannot be
described using the standard provided by classical probability
theory, but turns out to be more consistent with QP theory.
That doesn’t make QP a good tool for rational analysis, but it
would make it an interesting psychological mechanism, particularly if it is possible to provide clear and consistent interpretations
for its central constructs. Should events unfold in this way, then
statistics would continue to rely on classical probability for its
theoretical foundation, but cognitive modelers could use
quantum probability in many instances. There is nothing incompatible about these two states.
A quantum of truth? Querying the alternative
benchmark for human cognition
doi:10.1017/S0140525X12003068
Ben R. Newell, Don van Ravenzwaaij, and Chris Donkin
School of Psychology, University of New South Wales, Sydney, 2052 NSW,
Australia.
[email protected]
[email protected]
[email protected]
http://www2.psy.unsw.edu.au/Users/BNewell
http://www.donvanravenzwaaij.com
http://www2.psy.unsw.edu.au/users/cdonkin
Abstract: We focus on two issues: (1) an unusual, counterintuitive
prediction that quantum probability (QP) theory appears to make
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regarding multiple sequential judgments, and (2) the extent to which
QP is an appropriate and comprehensive benchmark for assessing
judgment. These issues highlight how QP theory can fall prey to the
same problems of arbitrariness that Pothos & Busemeyer (P&B) discuss
as plaguing other models.
1. Multiple sequential judgments. One of the basic tenets of
quantum probability (QP) is that the order in which questions
are asked of a person will affect how he or she feels about the
answer. Pothos & Busemeyer (P&B) illustrate this sequential
nature of QP using the Clinton/Gore attitude assimilation effect
reported by Moore (2002). The key result is that the percentage
of participants endorsing Clinton as honest increases by 7%
when Clinton is rated after Gore, but Gore’s honesty endorsement
decreases by 8% when he is asked about after Clinton. Thus the
politicians become more similar (assimilate) when they are
asked about second (3% difference in endorsement rates) than
when asked about first (18% difference in endorsement rate).
This point is illustrated in Figure 3 of P&B, reprinted here as
the top-left panel of Figure 1.
P&B show that if the initial state vector is projected onto the |
Gore yes> basis vector first, followed by the |Clinton yes> basis
vector, Clinton will be judged as more honest than if the initial
state vector is projected onto |Clinton yes> directly. Thus, the
authors explain how asking about the honesty of Gore first, will
lead to a subsequently more positive judgment of Clinton’s
honesty.
An unusual prediction that follows is that as these projections
continue, the state vector will gravitate toward the zero point.
As an illustration, consider the effect of asking successive questions about the honesty of additional presidents. We assume
that subsequent questions have representations as basis vectors
in the outcome space. Just as the state vector from |Gore Yes>
is projected onto |Clinton Yes>, we assume that subsequent questions cause the state vector to project onto the next appropriate
basis vector. As shown in Figure 1, as each state vector projects
onto the nearest point of the next basis vector, subsequent state
vectors will get shorter (by definition).
Although we agree that asking about the honesty of a number
of politicians might put one in a progressively more suspicious
frame of mind, it seems unlikely that the believability of any
president should necessarily decrease (reaching close to zero
in as few as 10 questions) as more questions are asked.
Imagine, for example, if the sixth president was Lincoln or
Washington.
A possible solution to this problem is to assume that the state
vector somehow resets or recalibrates itself, perhaps because of
a decay of the effect of initial questions (i.e., forgetting). P&B
argue that one of the benefits of QP is that it is based on axiomatic
principles, thus avoiding problems of “arbitrariness” common in
other explanatory frameworks (e.g., heuristics). Adding a “recalibration” step would appear to be a post-hoc fix outside of the
main principles, and as such, something that P&B are at pains
to avoid. This example highlights why formal frameworks make
such attractive theoretical tools: they make strong, testable
predictions.
2. An appropriate benchmark? Two criteria have been prominent in the search for an appropriate benchmark for probability
judgment: correspondence and coherence (eg., Hammond
1996). These terms, stemming from philosophy, invite different
ways of assessing truth: via correspondence with observable
facts, and via having a set of internally consistent (coherent)
beliefs. Several commentators have argued that both criteria
need to be considered for adequate assessment of judgments
(e.g., Dunwoody 2009; Newell 2013).
P&B argue strongly that coherence should be assessed against
the axioms of QP not CP – hence allowing Linda to be more
likely a feminist bank teller than just a bank teller – but what of
correspondence? Consider the correspondence error that homicide is judged the more likely cause of death than suicide (e.g.,
Lichtenstein et al. 1978). Such a judgment is an error because it
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
Figure 1 (Newell et al.). Multiple sequential judgments lead to a belief state that comes ever closer to zero. See text for details.
does not correspond with the fact that there are more suicides per
capita than homicides. Such an “irrational” judgment emerges
from the same cognitive system as the Linda judgment and, therefore, should, according to P&B’s thesis, be explicable in the QP
framework. Our intuition is that QP theory would explain this
effect by constructing bases corresponding to representations of
death from suicide, death not from suicide, death from homicide,
and death not from homicide (in much the same way as bases are
constructed for happy and ∼happy in P&B’s Fig. 1). It might be
assumed that people’s initial state vector, because of something
akin to “availability,” is closer to the homicide basis vector than
the suicide vector. This would lead to a larger projection, and,
therefore, a judgment of higher probability of homicide than
suicide.
Assuming that it is possible to construct such a space, one may
ask what predictions QP theory would make were we to ask the
participants to sequentially judge the likelihood of both suicide
and homicide. To generate such predictions, however, we must
first know whether, for example, the two questions are compatible. We must also know whether the initial vector lies between
the homicide and suicide basis vectors, or between the homicide
and not suicide vectors, for example. Such decisions about the
parameters of the model influence the qualitative pattern that
QP theory will produce, for example, that compatibility will determine whether we expect the judgments to be invariant to the
order of the questions. Similarly, the location of the initial state
vector, for incompatible questions, will determine whether the
second judgment increases or decreases relative to when it was
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
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Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
judged first. Although not relevant to the current example, the
principles of entanglement and superposition have similar
effects on the qualitative pattern that QP theory predicts.
To call the decisions about such principles in QP theory “arbitrary” may be going too far – P&B provide intuition for when we
might expect some of these principles to hold (e.g., compatibility).
However, we argue that an understanding of these unique aspects
of QP theory, to the point that they are predictable, is a major issue
that needs addressing before QP theory can vie to be the framework of choice.
Quantum modeling of common sense
doi:10.1017/S0140525X1200307X
Hamid R. Noori and Rainer Spanagel
Institute of Psychopharmacology, Central Institute for Mental Health, Medical
Faculty Mannheim, University of Heidelberg, J 5, 68159 Mannheim, Germany.
[email protected]
[email protected]
http://www.zi-mannheim.de/psychopharmacology.html
Abstract: Quantum theory is a powerful framework for probabilistic
modeling of cognition. Strong empirical evidence suggests the contextand order-dependent representation of human judgment and decisionmaking processes, which falls beyond the scope of classical Bayesian
probability theories. However, considering behavior as the output of
underlying neurobiological processes, a fundamental question remains
unanswered: Is cognition a probabilistic process at all?
Using quantum theory for understanding cognitive processes was
mainly inspired by the Copenhagen interpretation of quantum
mechanics in the 1920s. However, it took scientists almost a
century to formalize cognitive models utilizing the unique features
of quantum probability (Aerts & Aerts 1995).
Based on the Kolmogorov probability axioms, classical theories
rely on intuitive mathematical foundations that have substantiated
their application for modeling psychological phenomena for a long
time. Nevertheless, a large body of evidence on order and context
dependent effects and the violation of the law of total probability
suggest that human judgment might follow counterintuitive
principles.
One of the major achievements of quantum theory was the conceptualization of the superposition principle. Measurements of a
system, which is linearly composed by a set of independent
states, assign it to a particular state and stop it from being in
any other one. As a consequence, the act of observation influences
the state of a phenomenon being observed, and indicates a general
dependency on the order of observations. These are two significant aspects of cognitive processes that dramatically challenge
the classical theories.
The lack of reliable mathematical formalisms within the classical frameworks to address such issues, the inclusion of the classical
assessments as specific (trivial) quantum assessments, and the
wider application range of quantum probability, even suggest
the superiority of quantum theory for modeling cognition.
Pothos & Busemeyer (P&B) discuss this matter thoroughly and
elegantly. Modeling non-commutative processes is a unique
characteristic of quantum theory, which provides a noticeable
distinction between compatible and incompatible questions for
cognitive systems. From the psychological point of view, incompatibility between questions refers to the inability of a cognitive
agent in formulating single thoughts for combinations of corresponding outcomes. Each question influences the human state
of mind in a context-dependent manner and, therefore, affects
the consideration of any subsequent question (induced order
dependency). Bayesian models simply fail to represent the
context- and order-dependent scenarios, whereas for non-dynamic
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compatible questions their predictions converge to the assessment
of the quantum probability theories.
In addition to the more general nature of quantum theory in
describing static cognitive processes, this formalism could also
be viewed as an extension of the classical probabilities for
dynamic processes. Whereas time evolution is in both theories
represented as linear transformations, the dynamic quantum
probabilities are nonlinear functions, which in general implicate
possible violations of the law of total probability.
In the long run, the framework of quantum modeling of cognitive
processes presents a generalization of the Bayesian theories, with a
deeper notion of uncertainty and natural approaches to problems.
Despite the convincing superiority of quantum theories with
respect to classical probability models, a fundamental question
is still open: Are cognitive processes governed by stochastic principles at all? To address this question, we will focus on the compatibility and expediency of stochastic approaches from the point of
view of neuroscience research, and omit any involvement in philosophical discussions on the nature of human judgment and
decision making.
Following Griffiths and colleagues (2010), the authors characterize probabilistic models of cognition as “top-down” or “functionfirst.” Furthermore, P&B espouse the philosophy that “neuroscience methods and computational bottom-up approaches are typically unable to provide much insight into the fundamental why and
how questions of cognitive process” (sect. 1.2) and hence suggest the
right modeling strategy as beginning with abstract (stochastic) principles and then reducing them into (deterministic) neural processes.
However, the assumption of a stochastic nature of cognition
and behavior has severe consequences both mathematically and
biologically.
1. Convergence of stochastic and deterministic results: Using a
deterministic dynamic systems model, computational neuroscience
and biophysics improved our understanding of neuronal processes
in the last decades. Particularly, the models reproduced and predicted neurochemical and electrophysiological processes that were
shown to induce alterations in the behavior of animals and human
(Knowlton et al. 2012; Maia & Frank 2011; Noori & Jäger 2010).
The sum of these theoretical models and their experimental validations suggests a deterministic relationship between neural processes and behavior. On the other hand, stochastic models of
cognition assign to each behavioral output a proper random variable
in a probability space. Therefore, a certain behavior could be characterized as a deterministic function of biological variations and a
random variable simultaneously. This paradoxical duality requires
the deterministic and stochastic functions to converge to the same
behavioral outcome under given conditions. Consequently, the compatibility of the top-down stochastic approaches with biological findings and the possibility of a reduction into lower-level neural
processes depend on the existence of appropriate convergence criteria, which have not been provided to date.
2. Interactions of neural processes and human behavior: From
cellular dynamics to oscillations at the neurocircuitry level,
numerous studies have identified biological processes that
define/influence behavior (Morrison & Baxter 2012; Noori et al.
2012; Shin & Liberzon 2010). Therefore, cognition is a causal consequence of a series of biological events. In light of these investigations, a cognitive process of a stochastic nature inherited its
“random” character from its underlying biology. In other words,
the top-down reduction of the abstract stochastic principles into
neural processes implies probabilistic dynamic behavior of
neural systems at different spatiotemporal scales. However, the
lack of theoretical or experimental models confirming the probabilistic nature of neural mechanisms challenges the proposed
top-down strategy.
In conclusion, although the quantum probability theories significantly extend the application field of classical Bayesian theories
for modeling cognitive processes, they do not address the general
criticisms towards top-down modeling approaches.
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
Quantum probability, intuition, and human
rationality
doi:10.1017/S0140525X12003081
Mike Oaksford
Department of Psychological Sciences, Birkbeck College, University of
London, London WC1E 7HX, United Kingdom.
[email protected]
http://www.bbk.ac.uk/psyc/staff/academic/moaksford
Abstract: This comment suggests that Pothos & Busmeyer (P&B) do not
provide an intuitive rational foundation for quantum probability (QP) theory
to parallel standard logic and classical probability (CP) theory. In particular,
the intuitive foundation for standard logic, which underpins CP, is the
elimination of contradictions – that is, believing p and not-p is bad.
Quantum logic, which underpins QP, explicitly denies non-contradiction,
which seems deeply counterintuitive for the macroscopic world about
which people must reason. I propose a possible resolution in situation theory.
One of the motivations underlying the psychology of human
reasoning has been the assessment of human rationality: that is,
comparing human reasoning performance with some standard
of rationality provided by a normative theory, be it standard bivalent logic, Bayesian probability theory, expected utility theory, or
quantum probability (QP) theory (Chater & Oaksford 2012).
Whether this is a proper function of psychological theories of
reasoning has been debated within the pages of Behavioral &
Brain Sciences (Elqayam & Evans 2011), but QP is a competitor
to classical probability (CP) theory precisely at the computational,
normative level. Pothos & Busmeyer (P&B) are quite explicit on
this point when they observe that under a QP interpretation,
the conjunction fallacy is not fallacious. This assertion makes
sense only with the understanding that it conforms to the dictates
of QP, considered as a normative theory of how people should
reason. However, P&B do not provide an account of why, intuitively, QP is rational. Rather, in section 5, “The rational mind,”
they concentrate on pointing out existing problems with the
claims of CP, to provide an account of rationality.
Although as P&B observe, there are problems for existing normative theories, there are very clear underlying intuitions about why
they are rational. For standard bivalent logic, the primary intuition
is that contradictions are bad – that is, it is irrational to believe
both p and not-p (e.g., chalk is white and chalk is not white). If
one follows the laws of logic, one will never fall into contradiction,
and, therefore, logic can provide a rational standard. Similarly for
probability theory, the primary intuition is that making bets that
one is bound to lose is bad. If one follows the laws of probability
in the Kolmogorov axioms, one will never make a bet one is
bound to lose, and therefore it can provide a rational standard.
P&B provide no similar intuitive understanding of how QP is
rational, even if this understanding is ultimately deficient in some
respects, as they point out for CP. Rather, P&B observe that QP
is loosely like bounded rationality, and is consistent with the isotropic
and the Quinean nature of human thought.
The lack of an intuitive grasp of rationality in QP relates to the
closing sentence of section 5: “For the real, noisy, confusing, everchanging, chaotic world, QP is the only system that works in
physics and, we strongly suspect, in psychology as well.” The
world they describe here is actually the microscopic world of
quantum events that supplies the underlying domain of QP –
that is, the world whose behaviour it is trying to describe.
However, the world that provides the appropriate domain for
logic and CP is a relatively stable macroscopic world of which
we have varying states of knowledge. Insofar as the statements
of these theories describe these worlds, they provide a theory of
what these statements mean. There are obvious complexities
here, but one requirement of a formalism is that, in Susan
Haack’s (1978) terminology, it should be capable of respecting
the appropriate depraved semantics (Haack 1978, p. 188) – that
is, people’s intuitive understanding of a domain.
Logic and CP, to the extent they have a rational foundation,
assume a world where events cannot both occur and not occur
and where there are objective values. The extent to which one
might be inclined to adopt QP is the extent to which one is willing
to view the macroscopic world as being much more like the microscopic world than we have so far considered, hence it can provide an
appropriate depraved semantics. One might also question whether
the actual world of everyday experience is the “noisy, confusing,
ever-changing, chaotic” place or whether this is the product of our
situatedness within in it. However we view it, one should not underestimate the magnitude of the change in the conception of the
macroscopic world underlying human cognition that QP entails.
For example, in quantum logic (Haack 1974), which underpins
QP in the same way that classical logic underpins CP, the law of
the excluded middle, or non-contradiction (not( p and not-p)),
does not hold – that is, the intuitive rational basis of classical
logic is explicitly denied. Therefore, in the microscopic world,
quantum events can occur and not occur. But to embrace
QP is – for some events at least, that is, when questions are
incompatible – to require people to reject non-contradiction
for some macroscopic events, which seems deeply counterintuitive. However, this may not be quite as counterintuitive as
it seems.
Although there is no scope to develop the connections here in
any detail, there is a semantic account more readily compatible
with QP. This account is provided by situation theory (see
Barwise & Perry 1983; for a recent review, see Stojanovic
2012), in which situations – construed as something between the
physical and the psychological, simplistically, the bit of world one
can see that can include other people who might have a different
perspective on the same situation – are the basic building blocks,
and objects and regularities only emerge as uniformities against
the background of the ever-changing situations in which we find
ourselves. Joining such a world view with QP may provide an intuitive, rational handle on QP. For example, perspectival terms, such as
“behind” (which are often assimilated to indexicals), reveal apparent
breakdowns of non-contradiction. Therefore, for example, “the table
is behind the sofa” may be true from my perspective in a situation,
but it may be false from yours. “Perspectival relativity” is an important feature of situation semantics.
In summary, whereas one cannot help but be impressed by the
empirical grasp of QP, its potential to provide an alternative
rational foundation for human reasoning requires further consideration, as for some events, the failure of non-contradiction
at the macroscopic level seems deeply counterintuitive.
However, there are accounts of the semantics of natural language
that appear compatible with QP, and those may provide a much
needed intuitive foundation.
What’s the predicted outcome? Explanatory
and predictive properties of the quantum
probability framework
doi:10.1017/S0140525X12003093
Timothy J. Pleskac, Peter D. Kvam, and Shuli Yu
Department of Psychology, Michigan State University, East Lansing,
MI 48824.
[email protected], [email protected], [email protected]
www.msu.edu/∼pleskact
Abstract: Quantum probability (QP) provides a new perspective for
cognitive science. However, one must be clear about the outcome the
QP model is predicting. We discuss this concern in reference to
modeling the subjective probabilities given by people as opposed to
modeling the choice proportions of people. These two models would
appear to have different cognitive assumptions.
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
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Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
What is sometimes required is not more data or more refined data
but a different conception of the problem.
—Roger Shepard (1987, p. 1318)
Shepard made this statement in arguing that a universal law of
generalization could be had only by formulating it around
stimuli existing in a psychological, not a physical, space. Only
then, according to Shepard, could principles of generalization
be established that were invariant to context. Pothos & Busemeyer
(P&B) are making a very similar argument. They contend that the
probabilistic nature of cognition may adhere not to the assumptions of classical probability (CP), but instead to the properties
of quantum probability (QP).
Perhaps this is the reconceptualization that is needed. Phenomena that were errors according to CP, such as violations of the conjunction rule, order effects, and violations of the sure-thing
principle, are consistent within the structural constraints of QP.
Moreover, a quantum account can offer a more parsimonious
explanation of the phenomena. Take, for example, the Linda
problem. The typical explanation of this problem has been the
representativeness heuristic in which people substitute similarity
for probability (Kahneman 2003). In comparison, similarity is
implicit in the QP calculation (reflected in the angle of the initial
state vector relative to the bases).
The QP model of the Linda problem and other judgments of
subjective probability (Busemeyer et al. 2011) are intriguing
because they describe how people actually assign probabilities.
However, P&B use the same QP modeling framework to describe
both the process of making subjective probabilities (e.g., the
Linda problem) and the process of making a choice (e.g.,
Clinton vs. Gore honesty choice). This failure to distinguish
between subjective probabilities and choice proportions presents
a concern. To illustrate, consider another version of the Linda
problem that researchers have used, in which people have to
identify which of two statements is more probable (e.g., “Linda
is a bank teller” or “Linda is a bank teller and is active in the feminist movement”) (e.g., Tversky & Kahneman 1983). When
applied to choice proportions, QP is used to predict the objective
probability of choosing either statement by projecting onto a basis
vector. However, in modeling subjective probabilities, the model
uses the same operator; therefore, it is as if people have direct
access to the probability of entering into that state. Is the same
cognitive system used for these different responses? To suppose
these two models are of the same class seems to be an error. To
do so would be to assume that subjective ratings of probability
are the same as the probability of choosing a particular response.
The distinction between modeling the probability distributions
over behaviors and modeling probability judgments is important.
In the Linda problem, the probability judgment response is only
one of many responses in which the conjunction fallacy is
observed. The primary data, for example, were in terms of
ranking eight possible hypotheses in order from least to most
likely (Tversky & Kahneman 1983). We also know that the
extent to which people commit the conjunction fallacy changes
based on the format of the response, such as with bets (Tversky
& Kahneman 1983), or with frequency formats (Mellers et al.
2001). It would seem that a more complete process level
account of the conjunction fallacy would explain how these different response formats (or measurements) work, and would inform
any model predicting the effect.
A more general comment regarding the thesis of the article is
that quantum theory can provide a “better” probabilistic framework for cognition. The flexibility of QP theory may become a
concern, as skeptics might argue that it lacks specific predictions.
However, many predictions may be overlooked simply because
they reflect common sense or because the theory is new enough
that due scrutiny has not been afforded to uncover psychological
inferences from Gleason’s theorem or the Hilbert space. Take, for
example, the prediction that projecting a state vector to a lower
dimensional subspace will result in lost amplitude. This does
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appear to be consistent with the famous Korea/China asymmetry
in similarity judgments (Tversky 1977), and certainly similar predictions can be made in other areas, such as comparing emotions
of different dimensionality.
Perhaps the more difficult aspect is explanation. This seems
tricky, as we are asked to take a different interpretation of a cognitive system under QP. For example, under a classical view, a
cognitive system, at any given time point, is in a particular state.
However, in QP we have superposition meaning a cognitive
system is in no state at all! This idea would appear to change
our ability to point to information as a causal mediating mechanism, as well as our understanding of the nature of mental events
themselves. It also seems to change the very nature of what it
means to explain cognition, as superposition seems almost uninterpretable. Hughes (1989) suggests that an explanation in
terms of QP is a structural explanation. That is, it shows how
the stochastic nature of cognition and its probability functions
can be modeled using Hilbert spaces.
Is a structural explanation sufficient for psychologists? This
concern and the larger set of issues it raises may not come as a particular surprise, as QP theory was originally designed only to
predict the probability of outcomes of physical events. The challenge of modeling subjective judgments and mental processes is
obviously a new challenge for QP, and, therefore, it may simply
be that more work is needed to attune the QP framework to
address psychological, rather than physical, problems. Even so,
we suspect that the success of QP in cognitive modeling will
depend largely on its reinterpretation, reapplication, and resulting
predictive power rather than its narrative explanation.
ACKNOWLEDGMENT
A grant from the National Science Foundation (0955410) supported
this work.
If quantum probability = classical probability +
bounded cognition; is this good, bad, or
unnecessary?
doi:10.1017/S0140525X12003172
Tim Rakow
Department of Psychology, University of Essex, Wivenhoe Park, Colchester
CO4 3SQ, United Kingdom.
[email protected]
http://www.essex.ac.uk/psychology/department/people/rakow.html
Abstract: Quantum probability models may supersede existing
probabilistic models because they account for behaviour inconsistent
with classical probability theory that are attributable to normal
limitations of cognition. This intriguing position, however, may overstate
weaknesses in classical probability theory by underestimating the role of
current knowledge states and may under-employ available knowledge
about the limitations of cognitive processes. In addition, flexibility in
model specification has risks for the use of quantum probability.
The case for using quantum probability that Pothos & Busemeyer
(P&B) present seems to be that quantum probability – like classical Bayesian probability – supports a probabilistic approach to
cognitive modelling, but – unlike classical probability – predicts
behaviour consistent with the attributes of bounded cognition
such as the normal limitations in memory, processing capacity,
and attentional control. However, before accepting this intriguing
proposal, there are questions to consider. Is classical probability
theory truly inconsistent with behavioural phenomena attributable to bounded cognition? Do we need alternatives to existing
models that already incorporate insights concerning bounded cognition? Are there dangers in a unified quantum probability framework that subsumes classical probability and bounded cognition?
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
One feature of assigning probabilities is that uncertainty can
be aleatory or epistemic. Aleatory uncertainty derives from randomness in determining outcomes, whereas epistemic uncertainty reflects incomplete knowledge of outcomes and their
precipitating factors. For example, prior to flipping a coin I
assign Prob(Heads) = 0.5 (aleatory uncertainty); however, I still
assign Prob(Heads) = 0.5 if the coin has been flipped but is
covered (epistemic uncertainty). Often, both kinds of uncertainty are implied in probability assessments: someone may be
uncertain whether he or she is employed at time t both
because there is some randomness in determining employment
termination, and because he or she does not know what managerial decisions have occurred. Importantly, from a classical probability perspective, probabilities are always conditional on
current knowledge; and, consequently, some of P&B’s illustrations are perhaps not so puzzling for classical probability
theory as they suggest. My state of knowledge does differ according to whether someone asks me if Gore is honest before or after
asking me whether Clinton is honest. In one case my knowledge
includes the information “I have just been asked to assess Clinton’s honesty,” and in the other case it does not.
Moreover, as per the standard explanations of order effects
that P&B acknowledge, not only does my assessment of Gore
alter when comparison with Clinton is encouraged, but also my
classification for “honesty” may change (e.g., the category boundaries for honesty move). If the two questions about Gore refer to
different events (i.e., alternative conceptions of honesty), there is
little puzzlement for classical probability theory when the
probabilities of “yes” differ between them. Therefore, if classical
probability theory informed by the bounded nature of cognition
can accommodate order effects (and other effects described by
P&B) what does it gain us to have an alternative – arguably less
intuitive – account from quantum probability theory? Numerous
probabilistic cognitive models (including those previously championed by P&B) already incorporate features of bounded
cognition, including limitations in memory (Bush & Mostellar
1955), attentional control (Birnbaum 2008), or processing
capacity (Tversky 1972). P&B seem to be proposing models
that bypass the need for instantiating definable psychological
processes within cognitive models, because quantum probability
can account for phenomena usually explained through bounded
cognition by another route. Seemingly, in quantum probability
models of cognition, all the “work” is done by a probability
theory with the flexibility to account for many findings, without
incorporating cognitive theory in the model. How does this
advance our understanding of psychological processes?
P&B propose that: “superposition appears an intuitive way to
characterize the fuzziness (the conflict, ambiguity, and ambivalence) of everyday thought” (sect. 1.1). Surely, we already have
an intuitive explanation for the fuzziness of everyday thought:
thinking includes difficult assessments of unpredictable events
(aleatory uncertainty), under conditions of incomplete knowledge
(epistemic uncertainty), where often only a subset of the potentially available information is used (bounded cognition). What
does “superposition” bring to the table if we already accept that
information is incomplete? How does “time evolution” differ
from acknowledging that knowledge states can change rapidly?
What does “interference” add to the simple observation that
attention shifts between the reasons or factors that underpin
assessments?
I acknowledge that it is a weak argument against quantum probability formulations of cognitive models that they offer no more
than existing accounts: there is no particular reason to disfavour
a framework just because other frameworks already exist, and
arguably there is always room for another account that addresses
a different level of explanation. In other words, if quantum probability = classical probability + bounded cognition, then quantum
probability theory deserves its place. That said, it seems odd to
propose a class of cognitive models that, if I understand P&B correctly, need fewer components that relate to cognitive processes
because quantum probability theory already predicts phenomena
that those cognitive processes explain.
Additionally, there are potential dangers associated with a framework that subsumes an earlier one. For compatible questions,
quantum probability reduces to classical probability, and, therefore,
P&B rightly assert that the modeler determines a priori which questions are (in) compatible (sect. 2.2). However, just how a priori is a
priori? If incompatibility is determined prior to specifying any
formal models but after summarising the data, we risk the following
unsatisfactory state of affairs:
1. How do we know that these questions are incompatible?
2. Because classical probability theory was violated.
3. Why does the model predict violations of classical probability
theory?
4. Because these questions are incompatible.
If incompatibility is determined prior to data collection, this
problem does not disappear, because we already know much
about predicting the kinds of situations in which order effects,
and other violations of classical probability theory, occur. If the
incompatibility of questions cannot be specified in some way
that is fully independent of the data, then quantum probability
models risk benefitting from the mother of all free parameters:
explaining data consistent with classical probability, and data
that violate classical probability theory. Similar concerns about
the flexibility in model specification afforded by quantum probability arise in the example of modeling the Linda problem.
The assumption behind placing the quantum probability subspaces was that feminism is “largely uninformative to whether a
person is a bank teller or not” (sect. 3.1). However, the standard
account that Linda is representative of a feminist but unrepresentative of a bank teller might imply that people perceive that feminism is informative (radical feminism being inconsistent with
bank telling). Therefore, for the legitimate application of
quantum probability to cognitive modeling, we may also need
clear guidance on the a priori specification of subspaces. Otherwise, without such checks and balances on the specification of
quantum probability models, although, seemingly, such models
could account for everything, they would explain nothing (Glöckner & Betsch 2011).
Quantum probability, choice in large worlds,
and the statistical structure of reality
doi:10.1017/S0140525X1200310X
Don Rossa and James Ladymanb
a
School of Economics, University of Cape Town, Cape Town, South Africa;
Department of Philosophy, University of Bristol, Bristol BS81TB, United
Kingdom.
[email protected]
http://uct.academia.edu/DonRoss
[email protected]
http://bristol.ac.uk/philosophy/department/staff/JL/jl.html
b
Abstract: Classical probability models of incentive response are
inadequate in “large worlds,” where the dimensions of relative risk and
the dimensions of similarity in outcome comparisons typically differ.
Quantum probability models for choice in large worlds may be
motivated pragmatically – there is no third theory – or metaphysically:
statistical processing in the brain adapts to the true scale-relative
structure of the universe.
Pothos & Busemeyer (P&B) propose quantum probability (QP)
theory as a general axiomatic structure for modeling incentivized
response (choice). Experiments in the Kahneman–Tversky tradition suggest that the relationship between information and
human behavioral choice does not respect the axioms of classical
probability (CP) theory, notably monotonicity. Furthermore,
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
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Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
sequences of choices violate the classical law of total probability.
Modelers have accommodated these departures from CP by
hypothesizing heuristics – procedures that are specially adapted
to specific types of situations – but in so doing postpone
developing a general, testable theory that formally integrates
aspects of cognition, for example, similarity judgment and
assessments of risk. The probability model developed by
quantum physicists is a formal alternative to CP that handles
many of the recalcitrant phenomena, and can reproduce
successful predictions of models based on classical probability
in the classical limit.
The premise that the CP-based theory of decision – which is
better characterized as a theory of incentive response – has been
refuted by experimental evidence is questionable. The mathematical modeling of risk and the econometric estimation of responses
to changes in structural parameters of risk models, are currently
enjoying technical innovation, starting with Hey and Orme
(1994) and surveyed in Harrison and Rutstrom (2008). There is
evidence that unintended constraints derived from less sophisticated risk models may be implicated in a range of phenomena
that have been thought to imply use of special-purpose heuristics
(e.g., Allais violations, loss aversion, hyperbolic discounting; see
Andersen et al. 2011; Harrison & Rutström 2009). This research
might be taken as supporting the model of Oaksford and Chater
(2007) on a descriptive rather than a normative interpretation,
with its ambition to underwrite optimization relativized to heterogeneous preference structures. As P&B note, the Bayesian
rationality model relies on objective value maximization rather
than subjective utility maximization to handle Dutch book problems, and this reliance is not acceptable to economists. Global
optimality of decision procedures is surrendered once one
allows that people’s risk preferences are heterogeneous and circumstance specific, and that utility functions are often rank
dependent. However, this undermines confidence in CP-based
Bayesian models only as accounts of decisions in what Binmore
(2009) calls “large worlds”; they may yet be the right models
for “small world” problems. The idea that any empirically adequate general theory of decision might exist for large worlds is
purely conjectural.
However, P&B’s main point about Bayesian rationality is more
interesting. CP derives some of its support from philosophers
and others imagining that thought should work well in large
worlds. Once we are wondering about large worlds, we are in
the territory of metaphysics. Suppose for the sake of argument
that P&B are right that QP offers the correct general theory of
human incentive response. This would invite the question as to
whether there is some general feature of the world that explains
why both fundamental physical structure and fundamental cognitive structure follow QP rather than CP. The more modest philosophy would be to regard QP in psychology as just a placeholder,
which we borrow from physics if we decide we should give up
on CP because, for now, it is the only formally worked out
alternative.
As P&B note, one possible motivation for the immodest philosophy is the hypothesis that cognition follows QP axioms because
the brain’s physical structure allows it to perform quantum computation. But this is mere speculation. Another suggestion is
that our view of CP as the default account results from a failure
of metaphysical imagination that has been observed in other
areas of science. The failure in question is the assumption that
there must be a general account of reality that applies to all
scales of structural complexity. This is a prejudice derived from
treating human mechanical interactions with medium-sized
stable “things” as the structural field we should universally
project. Many philosophers presume a kind of atomism that is
inconsistent with quantum physics (Ladyman & Ross 2007). Arguably, the philosophy underlying CP-based accounts of cognition is
what Russell (1921; 1918–1924/1956, pp. 177–343) called logical
atomism. Does denial of physical atomism put stress on logical
atomism, or this simply sloppy homology?
306
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
Ladyman and Ross (2007) explain and defend the scale relativity of ontology. By this we mean that the persistent patterns that
compress information and thus support generalizations and outof-sample predictions – in a word, existents – arise at different
scales that cross-classify reality and do not reduce to one
another. Conscious, deliberate theoretical reasoning of the kind
modeled by philosophers, we suggest, ignores scale relativity by
generalizing the kind of atomistic decomposition that generally
works well in everyday mechanical manipulations. P&B’s hypothesis suggests an interesting twist. Most cognition is not deliberate
theorizing; it is statistical data processing by neural networks, converging on relatively stable perceptions and expectations by drift
diffusion rather than deduction. If the world does not parse
neatly into components on a single scale, why should nervous
systems evolved to compress and store information about that
world incorporate a false restriction to the effect that all relations
of similarity and difference, understood in terms of structural
distance, be representable on a single scale? This perspective
helps us understand the difference between the small words
where Bayesian reasoning works and the large worlds where it
does not: a small world is a restriction to one scale.
Challenges remain for the application of QP to cognition. The
formalism and the physics may not come apart as neatly as P&B
suggest. Quantum mechanics has a well-confirmed law of time
evolution, the Schrödinger equation, governing systems at least
when they are not being measured. P&B offer no analogous law
of time evolution of cognitive states. Schrödinger time evolution
is based on the assignment of a Hamiltonian representing the
energy of the system, and we lack any analogue of energy as
well. Quantum mechanics involves effects where Plank’s constant – the quantum of action – is not negligible and the limit in
which the predictions of classical mechanics are reached is that
in which h goes to zero. There is no analogously well-defined
limit in the application to cognition.
Physics envy: Trying to fit a square peg into
a round hole
doi:10.1017/S0140525X12003160
James Shanteaua and David J. Weissb
a
Department of Psychology, Bluemont Hall, Kansas State University,
Manhattan, KS 66506; bDepartment of Psychology, California State University,
Los Angeles, CA 90032.
[email protected]
[email protected]
www.ksu.edu/psych/cws
www.davidjweiss.com
Abstract: Pothos & Busemeyer (P&B) argue that classical probability (CP)
fails to describe human decision processes accurately and should be
supplanted by quantum probability. We accept the premise, but reject
P&B’s conclusion. CP is a prescriptive framework that has inspired a
great deal of valuable research. Also, because CP is used across the
sciences, it is a cornerstone of interdisciplinary collaboration.
Although Pothos & Busemeyer (P&B) make a noteworthy effort
to apply quantum probability (QP) theory to what they term classical (Bayesian) probability (CP) theory, we feel that their effort
has fallen short on three grounds.
1. CP as a straw man. First, they cast CP as a theory of the mind.
For Ward Edwards, the founder of research on behavioral decision
making and the first to use the term “subjective probability,” as well
as for Kahneman & Tversky and their adherents, CP served as a
point of comparison. Although one might argue that sensible decisions
ought to follow CP, judgment and decision making (JDM) research
has largely been about how decisions made by people do not. To
use Baron’s (2004) terminology, CP may be useful as a prescriptive
theory of behavior, but not as a descriptive theory.
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
The appeal of CP to most psychologists is quite different. We
seek a way to describe uncertainty. Uncertainty is a central construct that all sciences need to address. If there is to be any
hope of communicating across disciplines (or subfields of psychology), we need to use that common construct.
standard against which to evaluate theories in psychology. The issue,
according to Lo and Mueller, is that “human behavior is not nearly
as stable as and predictable as physical phenomena” (Lo & Mueller
2010, p. 13). Therefore, tools appropriate for advances in physics
may not be nearly as useful in psychology.
2. Distorting behavior to accommodate mathematical
requirements. Second, in order to define probabilities, P&B
readily dichotomize a continuum. For example, they discuss two
questions: (1) Is the person happy? (2) Is the person employed?
They treat these questions as if they are equivalent, both having
yes-no answers, and assign probabilities to the responses. The
focus is on how the answers interplay: happiness might be more
probable if a person is employed (because he or she has money)
or less probable (because she hates her job).
However, happiness is a continuum, not a dichotomy. Translating from mathematical exercises into real research questions
allows this distinction to be seen clearly. For example, one
might try a manipulation to change a person’s degree of happiness. The change could be measured, perhaps, by asking, “How
happy are you now?” at various times. The amount of change, as
opposed to whether there has been a change of status, is
important.
Dichotomizing not only loses information (MacCallum et al.
2002), it can seriously distort substantive inferences. The latter
danger is illustrated in the critique of a respected index of clinically significant change (Jacobson et al. 1999) by Weiss et al.
(2009). In contrast, employment has only two, or maybe a few
(part-time, or multiple-jobs) states. Therefore, it makes sense to
use dichotomized responses in asking a subject about employment
(“did the manipulation change your employment status?”).
3. Complexity beyond the needs of the research domain. Third,
although the level of mathematical sophistication underlying QP is
admirable, we view it as a case of overkill. As argued by fourteenth
century logician and Franciscan friar William of Ockham, “entities
should not be multiplied unnecessarily.” This recommendation is
now commonly known as “Occam’s Razor.” Ultimately, of course,
whether complexity is excessive is a subjective matter. It is our
opinion that P&B have developed an unnecessarily complicated
alternative to CP.
Traditional CP is notable for its simplicity. Augmented with a
Bayesian perspective, it has been a remarkably valuable tool for
stimulating JDM researchers ranging from Edwards in the
1950s and 1960s, to Kahneman and Tversky in the 1970s and
1980s, to Gigerenzer and colleagues in the 1990s and 2000s, to
the recent Behavioral & Brain Sciences article by Elqayam and
Evans (2011) arguing for a distinction between normative
systems and competence theories (see Brase & Shanteau 2011).
Young (1996) explained to the Psychometric Society why he was
no longer interested in building increasingly more sophisticated
algorithms for multi-dimensional scaling (MDS). Young had
been one of the pioneers in developing MDS tools. His arguments
can be summarized by the following quotes from his presentation:
“Application to problems in Psychology and Social Sciences have
gotten lost in the process of developing more complex methods
and models… The methods and models have gotten much
more complicated than users can understand…We are too
focused on mathematics at the expense of usefulness.” We
believe that Young’s comments on MDS are equally applicable
to the article by P&B.
4. Physics Envy. There has been ongoing discussion among behavioral scientists regarding “Physics Envy” (Mirowski 1992;
Sapolsky 1997; Schabas 1993). In part, recent discussions in economics (Lo & Mueller 2010) have been motivated by the role of socalled “quants” in the 2007 financial meltdown. As Malkiel noted,
whereas “physical models can provide an accurate description of
reality…financial models, despite their mathematical sophistication, can at best provide a vast oversimplification of reality”
(Malkiel 2011, p. 1)
Although we are not as pessimistic about the role of mathematical
models in psychology, we share the concern about using physics as a
Realistic neurons can compute the operations
needed by quantum probability theory and
other vector symbolic architectures
doi:10.1017/S0140525X12003111
Terrence C. Stewart and Chris Eliasmith
Centre for Theoretical Neuroscience, University of Waterloo, Waterloo, ON N2L
3G1, Canada.
[email protected]
[email protected]
http://ctn.uwaterloo.ca/
Abstract: Quantum probability (QP) theory can be seen as a type of vector
symbolic architecture (VSA): mental states are vectors storing structured
information and manipulated using algebraic operations. Furthermore,
the operations needed by QP match those in other VSAs. This allows
existing biologically realistic neural models to be adapted to provide a
mechanistic explanation of the cognitive phenomena described in the
target article by Pothos & Busemeyer (P&B).
If we are to interpret quantum probability (QP) theory as a
mechanistic cognitive theory, there must be some method
whereby the operations postulated by QP are implemented
within the human brain. Whereas this initially seems like it
would require some sort of large-scale quantum effect in the
brain, the target article notes that “the relevant mathematics is
simple and mostly based on geometry and linear algebra” (sect.
1.2). No special quantum physics effects are needed. If we can
show how neurons can compute these operations, then we can
interpret QP as making strong claims about how brains reason,
rather than merely acting as a novel behavioral description of
the results of cognitive processing.
Interestingly, there is already a family of cognitive models that
make use of geometry and linear algebra to describe cognitive
mechanisms, and these share many features with QP. Vector symbolic architectures (VSAs; Gayler 2003) use high-dimensional
vectors to store structured information, and use algebraic operations to manipulate these representations. The closest match to
QP among VSAs is holographic reduced representations (HRRs;
Plate 2003). As with QP, vectors are added to combine information and the dot product is used to evaluate similarity. For
example, if there is one vector for HAPPY and another for
UNHAPPY, the current mental state representation might be
0.86HAPPY+0.5UNHAPPY, representing a state more similar
to HAPPY than to UNHAPPY. While the notation is different,
this is identical to the |Ψ> = a|happy> + b|unhappy> example in
the target article (sect. 2.1; Fig. 1a). If EMPLOYED and UNEMPLOYED are other vectors that are similar to HAPPY and
UNHAPPY, respectively, then we get Figure 1b.
For the case of Figure 1c, HRRs and QP differ. Rather than
using tensor products like |happy>⊗|employed> (sect 2.2.2),
HRRs use circular convolution (HAPPY⊛ EMPLOYED). This
was specifically introduced by Plate as a compressed tensor
product: an operation that gives the same effects as a tensor
product, but that does not lead to an increase in dimensionality.
To explain this, consider the transition from Figure 1b to 1c in
the target article. In the first case, we are dealing with a twodimensional space, and in the second it is a four-dimensional
space. How is this represented in the brain? How do neurons
dynamically cope with changing dimensionality? Are different
neurons used for different cases? For HRRs, these concerns are
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
307
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
addressed by having a fixed (but large) dimensionality for all representations. If HAPPY is a particular 500-dimensional vector,
and EMPLOYED is a different 500-dimensional vector, then
HAPPY⊛EMPLOYED gives a new 500-dimensional vector (a
tensor product would give a 250,000-dimensional vector). Importantly, in high-dimensional spaces, the newly created vector is
highly likely to be (almost) orthogonal to the original vectors.
This gives a close approximation to all of the required orthogonality requirements mentioned in the target article, but does not lead
to an unlimited explosion of dimensions as representations get
more complicated. As we have shown elsewhere, adult human
vocabularies fit well within 500 dimensional spaces (Eliasmith,
in press).
Given cognitive theories expressed in terms of vector symbolic
architectures, we have created large-scale neural models that
implement those theories. In particular, we use the neural engineering framework (NEF; Eliasmith & Anderson 2003), which
gives a principled method for determining how realistic spiking
neurons can represent vectors, how connections between
neurons can implement computations on those vectors, and how
recurrent connections can be used to provide memory and
other dynamics. This allows us to turn abstract descriptions of cognitive processing into specific brain mechanisms, connecting a
plethora of neural data (functional magnetic resonance imaging
[fMRI], electroencephalograms [EEG], single cell recordings)
to cognitive function.
In the NEF, distributed representations of vectors are made by
generalizing the standard notion of each neuron having a particular preferred direction vector (e.g., Georgopoulos et al. 1986).
Whereas Hebbian learning rules can be used to adjust connection
weights, we can also directly solve for the desired connection
weights, as this kind of distributed representation allows a much
larger range of functions to be computed in a single layer of connections than is seen in typical connectionist models. This makes it
straightforward to create models that accurately compute linear
operations (such as the dot product), and even more complex
functions such as a full 500-dimensional circular convolution.
These models are robust to neuron death and exhibit realistic
variability in spiking behavior, tuning curves, and other neural
properties.
Although these techniques have not yet been used on the
specific tasks and theories presented in the target article, all of
the operations mentioned in the article have been implemented
and scaled up to human-sized vocabularies (e.g., Eliasmith
2005; Stewart et al. 2011). Furthermore, we have shown how to
organize a neural control structure around these components
(based on the cortex–basal ganglia–thalamus loop) so as to
control the use of these components (e.g., Eliasmith, in press;
Stewart & Eliasmith 2011). This architecture can be used to
control the process of first projecting the current state onto one
vector (HAPPY) and then on to another (EMPLOYED),
before sending the result to the motor system to produce
an output. These neural models generate response timing predictions with no parameter tuning (e.g., Stewart et al. 2010),
and show how the neural implementation affects overall behavior. For example, the neural approximation of vector normalization explains human behavior on list memory tasks better
than the ideal mathematical normalization (Choo & Eliasmith
2010).
Although the NEF provides a neural mechanism for all of
the models discussed in the target article, it should be noted
that this approach does not require Gleason’s theorem, a core
assumption of QP (sect. 4.3). That is, in our neural implementations, the probability of deciding one is HAPPY can be dependent not only on the length of the projection of the internal state
and the ideal HAPPY vector, but also on the lengths of the other
competing vectors, the number of neurons involved in the representations, and their neural properties, all while maintaining
the core behavioral results. Resolving this ambiguity will be a
key test of QP.
308
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
Why quantum probability does not explain the
conjunction fallacy
doi:10.1017/S0140525X12003123
Katya Tentoria and Vincenzo Crupib
a
Center for Mind/Brain Sciences (CIMeC), Department of Psychology and
Cognitive Sciences (PSC), University of Trento, Palazzo Fedrigotti, Corso
Bettini, n. 31, 38068 Rovereto, Italy; bDipartimento di Filosofia e Scienze,
dell’Educazione, Via S. Ottavio 20, 10124 Torino, Italy.
[email protected]
[email protected]
http://www.unitn.it/en/cimec/11750/katya-tentori
Abstract: We agree with Pothos & Busemeyer (P&B) that formal tools can
be fruitfully employed to model human judgment under uncertainty,
including well-known departures from principles of classical probability.
However, existing findings either contradict P&B’s quantum probability
approach or support it to a limited extent. The conjunction fallacy serves
as a key illustration of both kinds of problems.
Pothos & Busemeyer’s (P&B’s) argument in favor of a quantum
probability (QP) approach to cognitive modeling relies on the following premises:
1. A large amount of empirical findings accumulated in the last
40 years shows that human judgment often violates key aspects of
classical probability (CP) theory.
2. Heuristic-based approaches, although interesting, are often
limited in their applicability and explanatory scope.
3. It is possible to model probability judgment on the basis of
formal tools, and use these to re-express and sharpen popular
heuristics.
We agree with P&B on all the mentioned assumptions.
However, we depart from P&B in our assessment of the potential
of their QP approach for achieving a better understanding of
human judgments under uncertainty. We will illustrate our perspective with reference to the conjunction fallacy (CF) (Tversky
& Kahneman 1982; 1983). The CF plays a key role in P&B’s argument because P&B claim that this prominent violation of CP laws
has a natural and straightforward explanation in their QP
approach. In what follows, we will illustrate two main problems
that arise with regard to P&B’s interpretation of the CF results.
The first problem is that the QP approach is contradicted by
empirical data. To begin with, it is unable to accommodate double
conjunction fallacies (e.g., the mile run scenario in Tversky & Kahneman 1983, p. 306) – that is, all those situations in which h1∧h2 is
ranked over each of h1 and h2 appearing in isolation (Busemeyer
et al. 2011, p. 202). Several single conjunction fallacy results are
also demonstrably inconsistent with the QP approach. For example,
suppose that, given some evidence e, the most likely statement has
to be chosen among three, namely, a single hypothesis h1 and two
conjunctions, h1∧h2 and h1∧∼h2, as in the following scenario:
K. is a Russian woman.
Which of the following hypotheses
do you think is the most probable?
– K. lives in New York.
– K. lives in New York and is an interpreter.
– K. lives in New York and is not an interpreter.
[e]
[h1]
[h1∧h2]
[h1∧∼h2]
Tentori et al. (2013) observed that P(h2|e∧h1) < P(∼h2 |e∧h1)
for the majority of participants. One can just as safely assume
that P(h2|e) < P(∼h2|e), as clearly only a tiny fraction of Russian
women are interpreters. On these assumptions, the QP account
of the conjunction fallacy demonstrably predicts that the judged
probability of h1∧h2 must be lower than that of h1∧∼h2 (see
Fig. 1), and, therefore, that fallacious choices for h1∧h2 must be
less than those for h1∧∼h2.
However, in contrast to this prediction, a significant majority
(70%) of the fallacious responses in the Russian woman scenario
Commentary/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
Figure 1 (Tentori & Crupi). A QP representation of the Russian woman scenario. To simplify notation, P is taken to already encode the
evidence e (Russian). In line with participants’ judgment, the basis vectors are displayed as to imply P(h2|h1) < P(∼h2|h1), whereas the
position of the state vector implies P(h2) < P(∼h2), as clearly only a tiny fraction of Russian women are interpreters. Under these
conditions, if we assume – as it seems plausible – that P(h1) ≥ P(h2), the QP approach does not allow for the conjunctive probability
of h1 and h2 to rank higher than the single conjunct h1. However, the QP approach yields the wrong prediction even if we assume
P(h1) < P(h2). As the figure illustrates, in this case, contrary to the participants’ judgment, it is the conjunctive probability of h1 and
∼h2, not that of h1 and h2, which must rank highest.
concerned h1∧h2 rather than h1∧∼h2. More generally, one can
prove that, for the QP account, if P(h2|e) ≤ P(h3|e) and P(h2|
e∧h1) ≤ P(h3|e∧h1), the judged probability of the conjunction
h1∧h2 on the assumption of e cannot be higher than that of
h1∧h3. In a series of four experimental studies (which employed
different elicitation procedures, experimental designs, classes of
problems, and content), Tentori et al. (2013) documented a
robust pattern of results inconsistent with this implication of the
QP model, as well as with other recent proposals, such as the averaging (Nilsson et al. 2009) and the random error (Costello 2009)
hypotheses, which imply that CF rates should rise as the perceived
probability of the added conjunct does. The results supported a
different explanation of the CF, based on the notion of inductive
confirmation (Crupi et al. 2008). Much like the QP approach,
this explanation relies on a well-defined formalism (Bayesian confirmation theory) while avoiding the limitations (e.g., post-hoc parameters) that P&B (sect. 4.1) ascribe to other Bayesian models.
The second problem is that, even when logically consistent with
the empirical data, P&B’s treatment nonetheless receives limited
support. The QP modeler is typically left with a number of choices
that are unconstrained by the model itself. Lacking independent
and clearly defined empirical input, the modeling exercise does
not achieve explanatory relevance. The Linda scenario serves as
an illustration. For their QP approach to account for the reported
judgment P(bank teller ∧ feminist) > P(bank teller), P&B need to
make various assumptions (sect. 3.1) on the angle between the
basis vectors, as well as on the position of the state vector.
Some of these assumptions are uncontroversial. Others are
quite subtle, however, and have non-trivial consequences. As
the left column of Figure 2 shows, keeping basis vectors equal,
a small shift in the position of the state vector is enough to
reverse the predicted ranking between P(bank teller ∧ feminist)
and P(bank teller) even if the perceived probability of the feminist
conjunct remains much higher than that of bank teller.
A similar situation arises with the Scandinavia scenario (Tentori
et al. 2004):
Suppose we choose at random an individual from
the Scandinavian population.
Which do you think is the most probable?
– The individual has blond hair.
– The individual has blond hair and blue eyes.
– The individual has blond hair and does not
have blue eyes.
[e]
[h1]
[h1∧h2]
[h1∧∼h2]
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
309
Response/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
Figure 2 (Tentori & Crupi). Two different plausible QP representations of the Linda and Scandinavia scenarios. The positioning of the
vectors in the top half is compatible with the observed conjunction fallacy judgment P(h1∧h2) > P(h1), whereas that in the bottom half is not.
Does the QP approach predict the observed prevalence of fallacious choices for h1∧h2? This crucially depends on how we
determine the vector space.
The right column of Figure 2 shows two different representations of the Scandinavia scenario (the state vector
and the h2 basis vector are simply switched). The QP
approach allows for both, and both appear reasonable.
However, yet again, in the two cases, opposite orderings of
P(h1∧h2) and P(h1) follow from the QP approach. For an
observed judgment to be taken as properly supporting the
QP explanation of the CF, the corresponding vector space
representation needs to be constrained on independently
motivated empirical grounds. Otherwise, we can only say
that the QP approach can be made compatible with some
(and not all) of the CF data. However, for a putatively comprehensive theoretical framework, being able to accommodate some empirical results does not equal predicting and
explaining a phenomenon.
ACKNOWLEDGMENTS
Research relevant to this work has been supported by the MIUR grant
“Problem solving and decision making: Logical, psychological and
neuroscientific
aspects
within
criminal
justice”
(PRIN,
n.2010RP5RNM_006) and by Grant CR 409/1-1 from the Deutsche
Forshungsgemeinshaft (DFG) as part of the prority program New
Frameworks of Rationality (SPP 1516).
310
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
Authors’ Response
Quantum principles in psychology: The
debate, the evidence, and the future
doi:10.1017/S0140525X12003226
Emmanuel M. Pothosa and Jerome R. Busemeyerb
a
Department of Psychology, City University London, London EC1V 0HB,
United Kingdom; bDepartment of Psychological and Brain Sciences, Indiana
University, Bloomington, IN 47405.
[email protected]
[email protected]
http://www.staff.city.ac.uk/∼sbbh932/
http://mypage.iu.edu/∼jbusemey/home.html
Abstract: The attempt to employ quantum principles for modeling
cognition has enabled the introduction of several new concepts in
psychology, such as the uncertainty principle, incompatibility,
entanglement, and superposition. For many commentators, this is
an exciting opportunity to question existing formal frameworks
(notably classical probability theory) and explore what is to be
gained by employing these novel conceptual tools. This is not to
say that major empirical challenges are not there. For example,
Response/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
can we definitely prove the necessity for quantum, as opposed to
classical, models? Can the distinction between compatibility and
incompatibility inform our understanding of differences between
human and nonhuman cognition? Are quantum models less
constrained than classical ones? Does incompatibility arise as a
limitation, to avoid the requirements from the principle of
unicity, or is it an inherent (or essential?) characteristic of
intelligent thought? For everyday judgments, do quantum
principles allow more accurate prediction than classical ones?
Some questions can be confidently addressed within existing
quantum models. A definitive resolution of others will have to
anticipate further work. What is clear is that the consideration of
quantum cognitive models has enabled a new focus on a range of
debates about fundamental aspects of cognition.
R1. Beyond classical probability (CP) theory: The
potential of quantum theory in psychology
As we mentioned in our main article, quantum probability
theory simply refers to the theory for assigning probabilities
to events developed in quantum mechanics, without any of
the physics (cf. Aerts, Broekaert, Gabora, & Sozzo
[Aerts et al.]). We are interested in whether the computational properties of quantum models provide a natural
way to describe cognitive processes. Quantum cognitive
models do not require or assume that any aspect of brain
neurophysiology is quantum mechanical. The uniqueness
of quantum approaches relative to classical vector models
has more to do with their use of projections on subspaces
to compute probabilities rather than the fact that
complex vector spaces are employed (Behme). Some
quantum cognitive models can be realized with only real
vector spaces, but others need to use complex vector
spaces. The key features of quantum theory arise from
certain mathematical theorems, such as Heisenberg’s
uncertainty principle for incompatible observables, Gleason’s theorem (establishing the necessity for computing
probabilities from the squared length of projections) and
the Kochen–Specker theorem (establishing the necessity
for superposition states within a vector space). Because of
such theorems, we have properties such as incompatibility
and entanglement, which have no analogues in CP theory
or classical vector space theory. It is possible to have heuristic versions of such ideas, but a formal implementation
with a theory of probability is not possible, unless one
employs quantum probability theory. The need for
complex vector spaces in particular tends to arise when
the empirical result involves violations of the law of total
probability, as such violations indicate the presence of
interference terms. On a related subject, do quantum
models provide a unique insight about cognition? Notions
such as incompatibility, superposition, and entanglement
have been alien to cognitive psychology (the early relevant
insight of William James, which influenced the founding
fathers of quantum mechanics, had little impact on psychologists), until quantum cognitive models started to emerge.
These are all ideas that have no underpinning in any of the
other formal theories.
Lee & Vanpaemel question whether any insight about
cognition can be provided from quantum models. They
argue that “quantum theory assumes deterministic causes
do not exist, and that only incomplete probabilistic
expressions of knowledge are possible,” implying that classical theory does allow deterministic causes. We think this
distinction is wrong. If I know that Clinton is honest,
then there is a probability Probability (Gore|Clinton),
that Gore is honest, too. What determines the resolution
of my opinion that Gore is honest, given that Clinton is
honest? This is as mysterious in probabilistic models of cognition, as in quantum theory. The behavior we observe is
probabilistic; therefore, the models must generate predictions that are probabilities. Moreover, a sense of probabilistic determinism can arise in quantum theory in a way
analogous to that of classical theory: in quantum theory, if
it is likely that thinking that Gore is honest makes Clinton
likely to be honest too, then the subspaces for the corresponding outcomes are near to each other. It will not
always be the case that a person thinking that Gore is
honest will also think that Clinton is honest, but, on
average, this will be the case. The corresponding prediction/implication from classical theory would seem identical.
Also, the Schrödinger equation (which is used for dynamics
in quantum theory) is a deterministic differential equation,
analogous to the Kolmogorov forward equation (which is
used in stochastic models of decision making). In
quantum theory, the only source of non-determinism
comes from the reduction of the state vector.
Lee & Vanpaemel note that “effects of order, context,
and the other factors…might be hard to understand, but
adopting the quantum approach requires us to stop
trying.” On the contrary, quantum models provide a
formal mathematical framework within which to understand context and order effects in probabilistic judgment.
Such effects no longer have to be relegated to ad hoc
(e.g., order) parameters or conditionalizations; rather they
can be modeled in a principled way, through the relation
between the subspaces corresponding to different outcomes. A related point is that “discussion of entanglement…makes it impossible to construct a complete joint
distribution between two variables, and therefore impossible to model how they interact with each other.” The
first part of the sentence is correct, but not the second:
Entanglement does not mean that it is impossible to
model the interaction between two variables, but rather
that it is impossible to do so in a classical way. This is
because, for entangled variables, the interaction exceeds
classical boundaries. Contrary to what Lee & Vanpaemel
suggest, the objective of quantum cognitive models is
exactly to provide insight into those aspects of cognitive
process for which classical explanation breaks down.
Gelman & Betancourt note that, in social science, a
theory that allows the complete joint to change after each
measurement makes sense, and they outline some situations that might reveal violations of the law of total probability. These observations support the use of quantum
theory in corresponding models, although Gelman &
Betancourt also questioned some aspects of current
quantum cognitive models.
Pleskac, Kvam, & Yu (Pleskac et al.) also wondered
whether the notion of superposition might reduce insight
into quantum models. This is not the case. Note first that
a superposition state would have no specific value with
respect to a particular question; with respect to another
question, the same state may well have specific values.
Second, the fact that a superposition state has no specific
values (relative to a question) does not mean that this
state contains no information. On the contrary, the relation
of the state to the various subspaces of interest is the key
determining factor in any probabilistic estimate based on
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the state. In other words, a superposition state can still have
highly principled, well-defined relations, with respect to all
questions that we are considering (that is, all the relevant
subspaces in the Hilbert space in which the state is
defined).
The debate in the target article focused on the relation
between quantum and CP theory, because these are the
most comparable quantities. True, we criticize explanations
based on individual heuristics. This is because, if one adopts
a heuristics approach in cognitive modeling, there are few
constraints over what heuristics can be included and the
relation between heuristics. Inevitably, this endows a heuristics approach with considerable flexibility. By contrast, with
formal probability approaches (classical or quantum), before
one writes one line of his or her model, there are plenty of
relevant constraints., But, we do not wish to dismiss heuristics
(Khalil); rather, we prefer to explore whether heuristics can
be interpreted within a more formal framework. Also, we certainly do not dismiss bounded rationality. Indeed, this idea is
one of the main ways to motivate a perspective of quantum, as
opposed to classical, rationality.
Gonzalez & Lebiere rightly point out that cognitive
architectures, such as Adaptive Character of Thought –
Rational (ACT-R), go some way toward addressing our criticisms of approaches based on individual heuristics. This is
because a cognitive architecture would have in place
several constraints to prevent (some) post hoc modifications. One advantage of cognitive architectures is that
their conceptual foundation is usually closer to intuition
about psychological process. For example, a cognitive
architecture could involve memory and association processes, which are obviously related to human cognition.
We agree that cognitive architectures have a high appeal
in relation to psychological explanation. One unique advantage of modeling on the basis of formal probability theory is
that all the elements of a corresponding model either have
to arise from the axioms of the theory (and so are tightly
interconnected) or have to correspond to straightforward
assumptions in relation to the specification of the modeling
problem. Arguably, this is why some of the most famous
empirical demonstrations in psychology concern demonstrations that human behavior is inconsistent with the
axioms of formal (classical) probability theory (such as
Tversky & Kahneman 1983), whereas corresponding
results in relation to cognitive architectures are less prominent. Overall, whereas we agree that models based on cognitive architectures are more appealing than ones based on
individual heuristics, we still contend that formal probability theory, classical or quantum, provides a more principled approach for psychological explanation. Note also
that, as described in Busemeyer and Bruza (2012), it is
possible to embed quantum probability theory into a
larger quantum information processing system, using concepts from quantum computing (Nielsen & Chuang
2000). Converging the merits of formal probability theory
with cognitive architectures certainly has a priori explanatory appeal.
R2. Misconceptions on limitations
Even given the broad description of the theory in the target
article, we were impressed that some commentators were
able to develop their own variations of quantum models.
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This is encouraging in relation to how accessible quantum
theory is to a general cognitive psychology audience.
Equally inevitably, there were some misperceptions.
At a general level, we assessed quantum models against
results that have been at the heart of the decision-making
literature for nearly three decades now and have had a profound impact on the development of the debate regarding
the application of formal probability theory to cognition.
Whether they are simplistic or not, they constitute the
first testing ground for any relevant theory (Behme).
Also, we do not use “physics as a standard against which
to evaluate theories in psychology” (Shanteau & Weiss).
We do not use physics, rather a scheme for assigning probabilities to events from physics. Also, we do not use
quantum theory as a standard. Rather, we employ it as an
alternative theory, in situations in which it looks as if classical theory failed.
Kaznatcheev & Shultz (cf. Franceschetti & Gire)
point out that “the quantum probability (QP) approach is
a modeling framework: it does not provide guidance in
designing experiments or generating testable predictions.”
Of course, the same can be said of CP theory, neural networks, or mental models. But, does this mean that QP (or
any of the other frameworks) do not provide guidance
for predictions? We do not think that this is the case. For
each modeling framework, a researcher is called to scrutinize the unique features of the framework, and so
explore how these unique features can lead to novel predictions. In the case of QP theory, there are properties such as
superposition, incompatibility, and entanglement, which
offer possibilities for cognitive modeling, which are
unique in relation to other frameworks, such as CP
theory and neural networks. Relatedly, one cannot just
adopt a general formal framework and expect predictions
about psychological performance to emerge. One needs
to develop specific models (taking advantage of the
unique characteristics of the framework) and such models
will incorporate assumptions about psychological process
that go beyond the prescriptions of the framework. The
value of the psychological model is then a function of the
relevance of the mathematical theory, as well as the plausibility and motivation of these additional assumptions. An
example is our assumption in the quantum model for the
conjunction fallacy (and related effects) that more likely
predicates are evaluated first. Is this assumption problematic (Kaznatcheev & Shultz)? All we require is that the
decision maker has some mechanism of providing an
ordinal ordering of overlap.
A key characteristic of quantum models is that they are
not constrained by the unicity principle because they
allow for incompatible questions. Perhaps then, QP
models can be partly reduced to models of bounded cognition (cf. Rakow)? It is true that we motivate properties of
quantum theory, notably incompatibility, partly by appeal
to the unrealistic demands from the principle of unicity.
In other words, we suggest that the principle of unicity is
an unrealistic requirement for cognitive representations,
because it is unlikely that we can develop complete joint
probability distributions for the information in our environment. This recommends representations that are not constrained by the principle of unicity, that is, incompatible
representations. But there are other motivations completely independent of considerations relating to the principle
of unicity. For example, processing one question plausibly
Response/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
interferes with knowledge about another; the available
empirical results strongly indicate this to be the case, at
least in some cases (e.g., the conjunction fallacy; Busemeyer et al. 2011). Two independently good reasons for
some action appear to interfere with each other (e.g., violations of the sure thing principle in the prisoner’s dilemma;
Pothos & Busemeyer 2009). The meaning of individual
constituents appears to determine that of a conceptual
combination in a way that shows evidence for entanglement
(Aerts & Sozzo 2011b; Bruza et al. 2008). All these lines of
evidence could indicate properties of information processing in the cognitive system, plausibly independent of processing limitations, and which are (still) inconsistent with a
classical probabilistic description of cognition.
Ross & Ladyman say that quantum cognitive models
“offer no analogous law of time evolution of cognitive
states.” However, Schrödinger’s equation does exactly
this. For example, we have assumed that time evolution
can correspond to a thought process, whereby the initial
state develops to a state, which reflects the outcome of
the thought process (e.g., Pothos & Busemeyer 2009; Trueblood & Busemeyer 2012; Wang & Busemeyer, in press).
In such quantum cognitive models, the Hamiltonian may
not represent the energy of the system, but it does represent its dynamical properties, as is appropriate. Even
though there is no corresponding Planck’s constant in cognitive models, Planck’s constant has a unit-scaling role in
physical models. Relatedly, Kaznatcheev & Shultz question the nature of time development in quantum cognitive
models since “given enough deliberation time a participant
will always return to a mental state indistinguishable from
the one before deliberation.” Whereas this is true, deliberation in quantum models typically reflects some sort of
ambivalence between, for example, various options in a
decision-making problem. Importantly, resolving this
deliberation (e.g., accepting an option as a solution to a
decision-making problem) typically involves a reduction
of the state vector to a basis state. Collapsing a state,
which is a superposition state (in relation to possible outcomes for a question or problem), to a corresponding
basis state is an irreversible process, which breaks the
periodicity in quantum time evolution. This is the assumption we made in all the quantum models just mentioned.
Equally, the specific characteristics of quantum evolution
in quantum models do not always map well onto cognitive
processes, and in some cases classical models appear more
successful (Busemeyer et al. 2006).
More specific issues arise in various suggestions from
commentators. Newell, van Ravenzwaaij, & Donkin
(Newell et al.) note in their example of multiple projections that “it seems unlikely that the believability of any president should necessarily decrease…as more questions are
asked…” We entirely agree, but what these commentators
are computing is
|PLincolnYes . . . PGoreYes PClintonYes c|2
which is
Prob(Clinton is honest ^ then Gore is honest ^ . . .
then Lincoln is honest)
The conjunction that an increasing number of United
States politicians are all honest surely approaches zero
and there is no tension with the classical analogue, which
would simply be
Prob(Clinton is honest ^ Gore is honest ^ . . .
Lincoln is honest)
Moreover, if the state vector were to reset itself, we
would no longer be computing a conjunction, but rather
a conditional probability. For example,
Prob(Lincoln is honest| Clinton is honest)
= PLincoln cClinton 2
where ψClinton is a normalized state vector in the Clinton
subspace (this is Luder’s law). How large this conditional
probability is depends on the relation between the
Lincoln and the Clinton subspaces. In this particular
case, we can probably assume that knowledge that
Clinton is honest is uninformative relative to knowledge
that Lincoln is honest. Rakow rightly points out that the
state knowledge should differ when assessing the premise
that Clinton is honest, after deciding that Gore is honest,
and vice versa. We completely agree. But there is still
tension with CP. Note first that Probability (Clinton
honest) ∗ Probability (Gore honest |Clinton honest) is the
probability of deciding that Clinton is honest and Gore is
honest. Consistently with Rakow, the conditional tells us
whether Gore is honest, taking into account the information that Clinton is honest, but, Probability (Clinton
honest) ∗ Probability (Gore honest |Clinton honest) = Probability (Clinton & Gore). As conjunction is commutative,
the classical approach cannot model relevant question
order effects, without extra post hoc order parameters.
R3. Empirical and theoretical extensions
Aerts et al. point out that it is not just QP that is relevant.
Rather, there many aspects of quantum theory that are
potentially relevant to the modeling of cognition. By QP
we do not imply a particular aspect of quantum theory,
but rather the entire formalism for how to assign probabilities to events. Having said this, in pursuing the application
of quantum theory to cognition, our starting point has been
the more basic elements of quantum theory. We agree with
Aerts et al. and the other commentators, who pointed out
aspects of quantum theory, which could suitably extend
existing models.
For example, Shanteau & Weiss note that happiness (in
the target article example) is a continuum, and, therefore,
should not be modeled in a binary way. We only use
binary values to make the explanations as simple as possible,
and it is straightforward to develop models for more continuous responses (see, e.g., Busemeyer et al. 2006, for an
application to confidence ratings). For example, our demonstrations could be extended in a way such that the question
of interest can have an answer along a continuum, rather
than a binary yes–no. Such extensions could then be used
to explore differences in performance depending on differences in response format, as Pleskac et al. suggest.
Another issue concerns the output of quantum models. For
example, when we evaluate Probability (bank teller) for Linda
in the conjunction fallacy problem, what exactly does this
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probability correspond to (Kaznatcheev & Shultz; Pleskac
et al.)? We can use a pure state in a Hilbert space to represent
the state of an individual, for example, after a particular experimental manipulation (such as reading Linda’s description in
Tversky & Kahneman’s 1983, experiment). From this pure
state, we can estimate the probability that the individual will
assign in relation to, for example, the bank teller property
for Linda. We make the reasonable assumption (common to
probabilistic models of cognition, whether classical or
quantum) that individual response probabilities translate to
sample proportions for choosing one option, as opposed to
others. Relatedly, it is not clear what is to be gained by using
mixed states, in situations in which the emphasis is on individual responses and sample proportions are considered only as
indicators of individual behavior (Kaznatcheev & Shultz).
By contrast, if the emphasis is on ensembles (e.g., how
the behavior of a whole group of people changes), then
perhaps mixed states are more appropriate (Franceschetti
& Gire). Mixed states combine classical and quantum
uncertainties. They concern an ensemble of related
elements, for example, particles in a particular physical
system or students in an undergraduate class. The representation for each element is a pure (perhaps superposition) state, but we do not really know how many elements
are in specific pure states. Therefore, the overall system is
described in terms of all the possible pure states, weighted
by their classical probability of occurrence. Such mixed
states appear particularly useful in situations in which it is
impossible to consider a psychological situation in terms
of individual elements. For example, mixed states appear
appropriate when modeling convergence of belief in a
classroom or classification of new instances in a categorization model. Extending quantum models in this way is an
exciting direction for future research.
MacLennan raises some interesting points in relation to
how the basis vectors corresponding to different questions
are determined/learned. Currently, we make the following
minimal, but reasonable assumptions. Our knowledge space
would be populated with several possible subspaces, corresponding to questions that relate to our world knowledge.
For example, we would in general have subspaces corresponding to properties such as being a feminist or a bank teller.
Reasonably, the creation of knowledge would involve the creation of new subspaces in our knowledge space. Is there a
sense in which the overall dimensionality of the knowledge
space would increase, to perhaps reflect an overall quantitative
increase in knowledge? Perhaps, and this would require tools
from quantum theory, which go beyond those employed in
most current models, such as Fock spaces (Aerts et al.).
Another issue is how novel information could alter the relation
between subspaces. Quantum theory provides a mechanism
for doing so, through unitary evolution, although how to
specify unitary matrices that can capture the impact of a particular learning experience is beyond current models (but see
Ch. 11 in Busemeyer & Bruza 2012, for some proposals). Such
extensions can be a basis for quantum models of learning as
well (de Castro), although some corresponding insights are
possible from current models also. For example, de Castro discusses the idea that learning can involve a process of forgetting.
In current quantum models, after a projection, the initial state
loses some of the original information (which is like forgetting),
but new insights can be acquired about the initial state (see also
Bruza et al. 2009, for the applicability of QP in the understanding of human memory).
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Some commentators (Love; Marewski & Hoffrage)
observed that incorporating process assumptions in
quantum models could potentially greatly increase the
scope and testability of the models. We agree and offer
some preliminary responses, which show promise in this
direction. First, one distinguishing characteristic between
quantum and classical models is that, in the former, for
incompatible questions, all operations have to be performed sequentially. This naturally makes computation
sequential. In some cases, we have had to make particular
assumptions regarding projection order, for example, in the
conjunction fallacy. This assumption does not follow from
the axiomatic structure of quantum theory, rather it is
required because there is no other way to go from the
basic assumptions of quantum theory to a mechanism
that can be used to predict behavioral results (cf. Kaznatcheev & Shultz). If one proposes that projection
order, as specified at the computational level of a
quantum model, maps onto a psychological time parameter
for the actual computations at the algorithmic level, then
projection order could be tested as a process aspect of
quantum models. Second, in CP models in decision
making (such as random walk/diffusion models) it is
common to employ a time development process (from
the forward Kolmogorov equation), but in this case, the
constraints at baseline (e.g., the law of total probability)
would also apply at any subsequent time. However, in
quantum models, time evolution (from Schroedinger’s
equation) can be quite different. The states at baseline
may obey the law of total probability, but after time evolution the law of total probability may break down. We
have fruitfully employed this approach in a variety of situations (Pothos & Busemeyer 2009; Trueblood & Busemeyer 2012; Wang & Busemeyer, in press). In such
quantum models, the time development of the initial
state into another one (for which the law of total probability
can break down) is necessarily interpreted as a process of
deliberation of the relevant evidence. Therefore, such
models do have a process component, although a lot
more work needs to be done along these lines. Finally,
Busemeyer et al. (2006) explored a random walk
quantum model for a decision-making task. Therefore,
there is some groundwork on how to employ quantum principles to implement process models proper.
Behme and Corr highlighted the issue of how quantum
cognitive models scale up to more realistic empirical situations. It is probably fair to say that both the application
of classical theory and quantum theory have been focused
on idealized, laboratory situations, because of the complexity of taking into account a full set of realistic constraints in
a principled way. An example of progress in this respect is
the model for behavior in the Prisoner’s Dilemma games of
Pothos and Busemeyer (2009) (see also Trueblood & Busemeyer 2012; Wang & Busemeyer, in press). In both the
classical and the quantum versions of this model, it was
possible to incorporate the idea of cognitive dissonance
(Festinger 1957), so that beliefs and actions would converge to each other. The weight of this process, in relation
to evaluating payoff, could not be predicted a priori, and,
therefore, was set with a free parameter. Therefore,
“more realistic” mechanisms can, in principle, be
implemented in both classical and quantum models.
There is still a question of whether these mechanisms
allow successful coverage of empirical data. One point is
Response/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
that for empirical data that violate the basic axioms of classical theory, even postulating additional realistic mechanisms (such as cognitive dissonance), is still insufficient.
Specifically in the area of preference formation, Kusev
& van Schaik suggest that preferences are often constructed in an ad hoc, content-dependent way and Corr
also highlights the potential impact of mood or other
emotional variables on preference formation for the same
option. At face value, incompatibility in quantum theory
can allow evaluation of probabilities or preferences in a perspective-dependent way. Corr also points out that introducing considerations relating to mood or emotion may lead
to “nonlinear dynamical effects.” Because probability in
quantum models is assessed by squaring the length of the
state vector, there is more potential for capturing any corresponding non-linearities in the dynamics of a psychological system, although clearly, whether this is actually the
case or not can only be assessed in the context of specific
models. The work of Pothos and Busemeyer (2009), Trueblood and Busemeyer (2012), and Wang and Busemeyer
(in press) takes advantage of non-linearities in the development of probabilities for particular events, to capture
results that are puzzling from a classical perspective.
A more general point regarding the scalability of
quantum versus classical models concerns the requirements from the principle of unicity for the latter. Classically, there is the demanding requirement that there is a
complete joint probability distribution for the outcomes
of all questions. This requirement means that every time
an additional question is included, the complete joint
needs to be augmented to take into account the event
and, also, all corresponding marginal probability distributions need be specified as well. Therefore, additional
questions greatly increase the complexity of the required
representation. By contrast, representations of incompatible questions in a quantum model can coexist in a low
dimensionality space. The introduction of further incompatible questions does not necessarily increase the dimensionality of the space and the subspaces for different questions
can be related to each other through a unitary operation.
Overall, representation of additional questions (for a
hypothetical system) in a quantum model implies lower
representational demands than in a classical one.
A promising new domain of application of quantum theory
is learning and memory (as discussed, see de Castro; Bruza
et al. 2009). Baldo also discusses how the main elements of
signal detection theory can be represented in a Hilbert
space. This is an interesting extension to current work with
quantum cognitive models. One advantage is that this provides an intuitive visual representation of some related
visual phenomena. An issue for future research would be
whether there are empirical findings in signal detection
theory, which necessitate a Hilbert space representation, for
example, in relation to violations of the law of total probability
or order/context effects. In the more general area of perception, there has been some work along such lines, by Conte
et al. (2009) and Atmanspacher and Filk (2010).
R4. Empirical challenges
Whether researchers accept the quantum framework as a
viable alternative to CP theory is partly an empirical
issue. We review some important empirical challenges, as
highlighted by the commentators.
Hampton points out the quantum model for the conjunction fallacy cannot be applied to understand the
guppy effect – a situation in which the conjunction is
more probable than both the individual constituents.
Note first that we agree that researchers should be
looking to understand similarity and decision-making processes within the same modeling framework (Shafir et al.
1990). However, the mechanics of projection in the Linda
problem are quite different to those for the guppy effect.
In the Linda problem, the initial state reasonably corresponds to the impression participants form after the
Linda story. Then, this initial state is assessed against
either the single property of bank teller or the conjunction
of bank teller and feminist properties. By contrast, in the
guppy problem, the initial state is a guppy, a pet fish. If
we were to adopt a representation analogous to the one
for the Linda problem, we would need an initial state that
exists within a subspace for pet fish, which corresponds to
the concept of guppy. Then, a guppy has maximum alignment with the pet fish subspace (probability = 1) and less
alignment with either the pet or the fish subspaces. In
other words, in the guppy case it makes no sense to consider
the conjunction of pet and fish properties for guppy (as in
the Linda case), because by definition a guppy is constructed to be a pet fish; the pet fish characterization for
guppy it tautological, rather than a conjunction. By contrast,
Linda is not by definition a bank teller and a feminist, rather
Linda corresponds to a particular state vector, related to all
these properties. Therefore, the guppy effect can be reproduced with the quantum model for the Linda problem,
albeit in a trivial way. A problem with Hampton’s model is
that, as with averaging models generally, it fails to account
for violations of independence, that is, findings such that
A1 + B1 > A2 + B1, but A1 + B2 < A2 + B2 (Miyamoto
et al. 1995). Finally, an alternative, not trivial, quantum
model for the guppy effect has been proposed by Aerts
(2009; Aerts & Gabora 2005). This model is different in
structure from the Linda model (for example, it relies on
Fock spaces, which generalize Hilbert spaces in a certain
way), nevertheless both models are based on the same fundamental quantum principles.
Tentori & Crupi discuss an interesting finding according
to which, for a Russian woman, Probability (NY and I) is
higher than Probability (NY and ∼I), whereby NY means
living in New York and I being an interpreter. They show
how an application of the quantum model for the conjunction
fallacy in this case appears to break down. However, their
demonstration is based on a two-dimensional representation;
but quantum models are not constrained to two dimensions.
The dimensionality of the representation will be determined
by the complexity of the relation between the corresponding
questions. Dimensionality is not a free parameter: we just
need to ask whether all the information in a problem can
have an adequate representation in, for example, two
versus three dimensions. Note that if one parameterizes
the angles between subspaces, then dimensionality can
increase the number of parameters. However, with most
current demonstrations, the focus has been on showing
whether a result impossible to predict with (typically) CP
principles can emerge from a quantum model.
Regarding the problem of the Russian interpreter living in
New York, the fact that a two-dimensional representation is
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inadequate can be seen by noting that the state vector,
assumed to be within the Russian subspace, has to be
very close to both the ∼I and ∼NY rays, as, clearly, both
of these are extremely unlikely properties for a Russian
woman. Moreover, the I and NY rays cannot be close to
each other, because knowing that a person is an I does
not really tell us very much about whether he/she lives in
New York (but equally it does not preclude that the
person is living in New York). One can satisfy these tricky
constraints (along with the simpler one that the state
vector has to be closer to the ∼NY ray than the ∼I one)
in a three-dimensional space. Note, finally, that the
quantum model for the conjunction fallacy assumes that
the more likely predicate is evaluated first. Then, it
emerges that Prob (I and then NY)>Prob (∼I and then
NY), as required. Figure R1A–C illustrates the representation, but it is also possible to verify the prediction directly.
For example, setting NY=[0.39, 0.92, 0.047], ∼NY = [0.087,
−0.087, 0.99], I = [−0.39, 0.92, 0.047], ∼I = [−0.087,
−0.087, 0.99], Russian = [0.029, 0.087, 0.99], we have that
Prob (I and then NY) = 0.0065 and Prob (∼I and then NY)
= 0.0043. We note that this is a toy demonstration, which is
meant to neither reproduce specific numerical values nor
cover all the relevant constraints of Tentori & Crupi’s
example (for these, a more general model would be
needed). Nevertheless, the demonstration does capture
their main result.
A key question is exactly how definitive is the evidence
we consider in favor of quantum theory. Kaznatcheev &
Shultz rightly point out that there are rarely clear-cut
answers. They discuss the case of Aerts and Sozzo’s
(2011b) demonstration of entanglement and conclude
that one cannot necessarily infer the need for quantum
theory, over and above classical theory. Likewise, violations
of the law of total probability are amenable to classical
explanations, for example, with appropriate conditionalizations on either side of a law of total probability decomposition. We agree that an empirically observed violation of
the law of total probability does not definitely prove the
inapplicability of CP. However, classical explanations
would rely on post hoc parameters or conditionalizations,
with unclear explanatory value. In quantum models,
because of the order and context dependence of probabilistic assessment, it is often the case that corresponding
empirical results can emerge naturally, (nearly) just from
a specification of the information in the problem (as in
Figure R1. A representation that can account for the main finding discussed by Tentori & Crupi, that a Russian woman is more likely to
be living in New York and be an interpreter, than be living in New and not be an interpreter. Part C shows a magnification of the central
part of B.
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Response/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
the model for the conjunction fallacy; Busemeyer et al.
2011). Relatedly, Shanteau & Weiss questioned the complexity of quantum cognitive models. On the contrary, for
the reasons just discussed, it seems clear that quantum
models for empirical results such as the conjunction
fallacy are actually simpler than matched classical ones.
Quantum models can involve both incompatible and
compatible questions, classical models only the latter. Are
quantum models more flexible than classical ones? Do
quantum models subsume classical ones (Houston &
Wiesner, Kaznatcheev & Shultz)? There are three
responses. First, in all the quantum models that have
been employed so far, quantum transition (unitary)
matrices need to obey double stochasticity, a constraint
not applicable to classical transition matrices. Second, as
Atmanspacher notes (cf. Atmanspacher & Roemer
2012), it is possible to test in a very general way whether
a quantum approach can apply to a given situation. Consider a particular question and the conjunction of this question with another one, assumed to be incompatible. If the
individual question fails to be compatible with the conjunction (i.e., there are order effects between the question and
the conjunction), then a Hilbert space representation is
precluded. Third, there is the law of reciprocity, according
to which the probability of transiting from one pure state to
another must equal the probability of transition in the
opposite direction. In other words, if we imagine a state
vector in a unidimensional subspace A, ψA, and one in a
unidimensional subspace B, ψB, then
PA cB 2 = PB cA 2
which means that
Probability(A|B) = Probability(B|A)
but in classical theory,
Probability(A|B) = Probability(B|A)
in general. Wang and Busemeyer (in press) have tested the law of
reciprocity in question order effects and found it to be upheld
with surprising accuracy. It is important to note, however, that
the prediction
Probability(A|B) = Probability(B|A)
only holds for rays and not for events described by higher
dimensional subspaces.
These are abstract arguments and do not necessarily bear
on the issue of the relative complexity between particular
models. Are particular quantum models more flexible
than matched classical ones? This is an empirical issue.
So far, there has been one corresponding examination, by
Busemeyer et al. (2012), as reviewed in the target article.
Even taking into account relative model complexity, a
Bayesian analysis still favored the quantum model over a
matched classical one. This is one examination for one particular quantum model, but this examination does provide
the only available evidence and this evidence does its
small bit toward undermining a claim that quantum
models are in general more flexible than matched classical
ones.
Rakow notes that perhaps a major source of flexibility of
quantum models arises from a choice of whether some
questions are treated as compatible or incompatible (see
also Navarro & Fuss; Newell et al.). This point is overstated. First, the default assumption is that questions
about distinct properties are incompatible. Noori & Spanagel provide a good expression of the relevant idea:
“Incompatibility between questions refers to the inability
of a cognitive agent in formulating single thoughts for combinations of corresponding outcomes.” Conversely, compatible properties are such that it is possible to combine them
in a single corresponding thought. It is usually possible to
empirically determine incompatibility between two questions, as conjunction is usually subject to order effects. In
the context of Gore, Clinton question order effects,
Khalil argues that the questions “involve symmetrical and
independent information. Information with regard to
each question can be evaluated without the appeal to the
other” to conclude that these questions cannot be incompatible. All these reasons are exactly why it is so puzzling that
empirically we do observe order effects in the Clinton,
Gore question. Crucially, the considerations Khalil provides are not really the ones that are relevant in determining incompatibility; rather, these would relate to the
presence of order effects.
Second, once incompatibility/compatibility is resolved, a
QP model is limited in terms of both the relation between
the various subspaces and the location of the state vector.
These relations can be often determined in a principled
way, via correlations between answers to relevant questions
and considering the participants’ frame of mind in the
study, and do not prejudge the accuracy of model predictions. For example, Tentori & Crupi note that variations
in the representation for the quantum model for the conjunction fallacy can provide predictions inconsistent with
empirical results. Our representation for this model
(Fig. 2 in the target article) was created to be consistent
with three constraints: Probability (∼bank teller) is as
high as possible; Probability (feminist) is as high as possible;
finally, the ray for the feminist question is relatively neutral
with respect to the bank teller, not bank teller questions,
but slightly closer to the not bank teller question. All
these three constraints are part of the original specification
of the problem. They are self-evident and there is no circularity or subtlety in creating a quantum representation for
the Linda problem that is consistent with these constraints.
Even though there is some flexibility in the exact angles,
any plausible quantum representation would need to be
consistent with these constraints. Tentori & Crupi’s variation in the bottom left hand side of their Figure 2 violates
the first constraint, and, therefore, their corresponding
conclusion is incorrect.
Exactly how precise the specification of subspaces and the
state vector is will depend on exactly how precise we require
the predictions to be. For example, in the conjunction fallacy
problem, if we were interested in predicting a specific probability for preferring the conjunction to the single predicate
statement, we could fit the angle between the state vector
and one of the subspaces, but this is less interesting,
because it is a problem of fitting a one parameter problem
to a one degree of freedom result. What we believe is
much more interesting is whether a range of sensible placements for the bank teller, feminist subspaces and the state
vector can lead to a conjunction fallacy.
BEHAVIORAL AND BRAIN SCIENCES (2013) 36:3
317
Response/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
The Linda fallacy is an excellent example with which to
demonstrate the quantum approach to decision making,
because the state vector and location of subspaces are
well constrained. By contrast, in the Scandinavia scenario
discussed by Tentori & Crupi, there are fewer such constraints. We agree with Tentori & Crupi that a particular
configuration can lead to a conjunction fallacy, but others
do not. The crucial point is that QP theory in general and
the quantum decision-making model of Busemeyer et al.
(2011) in particular can naturally produce conjunction fallacies. Furthermore there are numerous testable consequences that follow from this model that are independent
of parameters and dimensionality. By contrast, no CP
model can naturally produce a conjunction fallacy,
because a fundamental constraint in such models is
Prob(A) ≥ Prob(A ^ B)regardless of what events A, B are
employed. To follow from Tentori & Crupi’s related criticism, the predictions from quantum models are well constrained, to the extent that the corresponding empirical
problem is well specified as well.
Because quantum theory is a formal theory of probability, all the elements in quantum theory constrain each
other. As Stewart & Eliasmith note, it is possible to
create neurally plausible vector-based representations, in
which probability for the outcome of a question can
depend on a variety of factors, for example, the length of
projections from all competing vectors. Such schemes
perhaps motivate a vector-based approach to representation, without the constraints from Gleason’s theorem.
However, without Gleason’s theorem, there appears to be
no principled reason to choose among the various possibilities for how to generate decisions. Gleason’s theorem
exactly justifies employing one particular scheme for relating projection to decision outcome and, moreover, allows
projection to be interpreted in a highly rigorous framework
for probabilistic inference. Employing Gleason’s theorem
reduces arbitrary architectural flexibility in vector models.
The debate over which empirical tests can reveal a necessity for a quantum approach, as opposed to a classical one,
will likely go on for a while. Dzhafarov & Kujala discuss
the idea of the selectiveness of influences in psychology,
and draw parallels with empirical situations in physics
that can lead to violations of Bell’s or the CHSH
(Clauser-Horne-Shimony-Holt; e.g., Clauser & Horne
1974) inequalities. In physics, the latter has been taken as
evidence that superposition states exist and, therefore, as
support for a view of physical reality with quantum mechanics. Can analogous tests be employed, in relation to selective influences or otherwise, to argue for the relevance of
quantum theory in psychology, and how conclusive would
such tests be? Some theorists (Aerts & Sozzo 2011b;
Bruza et al. 2008) have explored situations that show the
entanglement of cognitive entities, for example, the meanings of constituent words in a conceptual combinations.
Others (Atmanspacher & Filk 2010) have considered
entanglement in perception. The focus of our work has
been on decision-making results, which reveal a consistency with quantum principles (e.g., incompatibility, interference) and away from classical ones, but in psychology,
and unlike in physics, there are challenges in any empirical
demonstration. A key problem is that the psychological
quantities of relevance are not always as sharply defined
as physical quantities (e.g., word meaning vs. momentum).
Another problem is that in physical demonstrations it is
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often possible to eliminate noise to a very high degree,
but this is much less so in psychological experiments.
Relatedly, MacLennan points out that, if we know the
relation between two incompatible subspaces (as we
must, if we are to build a well-specified quantum model),
then it is possible to quantify the lower limit of the
product of the uncertainties for the corresponding observables. This raises the possibility that lowering one uncertainty may lead to increases in the other, and vice versa.
This is a novel and exciting possibility, not only in terms
of novel predictions, but also in terms of, perhaps, tests
for whether two properties are incompatible or not.
Blutner & beim Graben are more ambitious, and ask
whether it can be demonstrated that, in some case,
“quantum probabilities are such a (virtual) conceptual
necessity.” Are there general properties of a system of
interest (an aspect of cognition), which reveal a necessity
for quantum probabilities? This is surely a very provocative
thesis! These commentators put forward the idea of
quantum dynamic frames. The main idea in quantum
dynamic frames is that a measurement (typically) changes
the system. When there is a system of this type, under
general conditions, quantum dynamic frames necessarily
lead to the structures for probabilistic assessment, as in
standard quantum theory. Therefore, the question of
whether QP theory is relevant in cognitive modeling or
not can be partly replaced with the more testable/specific
question of whether measurement can change a cognitive
state.
R5. Neural basis
It is possible to utilize quantum theory to build models of
neural activity. MacLennan suggests that the distribution
of neural activity across a region of cortex could be
described by a quantum wave function. Whereas this possibility is intriguing, one has to ask whether there is any evidence that such aspects of brain neurophysiology involve
quantum effects or not. Banerjee & Horwitz offer a preliminary perspective in this respect. They note that to study
“auditory–visual integration one would design unimodal
control tasks (factors) employing visual and auditory
stimuli separately, and examine the change of brain
responses during presentation of combined visual–auditory
stimuli.” This is effectively a factor-based approach, but one
which, in a classical analytical framework, does not take into
account potential interference effects. Rather, the standard
assumption is such that the brain regions engaged by each
unimodal task have an additive impact in relation to the multimodal task. Relatedly, Banerjee & Horwitz point out that,
when considering functional networks, it is often the case
that any “any alteration in even a single link results in alterations throughout the network.” This raises the possibility that
the function of nodes in a functional network cannot be
described by a joint probability function, so that states of
the network may exhibit entanglement-like effects. Clearly,
we think that such applications are very exciting (cf. Bruza
et al. 2009).
The issue of quantum neural processing, as described
previously, is distinct to that of the neural implementation
of quantum cognitive models. According to Mender, if a
quantum approach to cognition was tied to a neural/physical theory of implementation, then, perhaps, pairs of
Response/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
incompatible observables could be extracted from such a
theory directly. By contrast, in our current, mostly topdown, approach we have to infer incompatible observables in purely empirical ways. Moreover, in such a cognitive approach it is not really possible to specify an
analogue to Planck’s constant. How much does the
quantum cognition program suffer from the lack of a
clear theory of neural implementation? We believe not
very much. We seek to infer the computational (and possibly process) (Love; Marewski & Hoffrage) principles of
cognitive processing. Such principles would be consistent
with a variety of implementational approaches at the
neural level. Although ultimately, cognitive computation
and neural implementation have to constrain each other,
these constraints are often fairly loose and in actual practice do not typically impact on the specification of computational models. Mender is correct that, perhaps, neural
implementation considerations might help us with the
determination of pairs of incompatible properties, but
even just focusing on the fact that certain pairs of observables are incompatible, and information about their
relation (which can be inferred from general properties
of the observables), seems to provide us with considerable
explanatory power.
There is a deeper issue here of whether there is something about the computations involved in quantum cognitive models that precludes neural instantiation. Noori &
Spanagel argue that brain neurobiology is deterministic
and, moreover, that it deterministically specifies behavior.
Therefore, how can we reconcile neural/biological determinism with behavioral stochasticity? First, the work
reviewed in Noori & Spanagel’s commentary shows a
causal relationship between neurochemical changes in the
brain and behavior, which is, perhaps, unsurprising, but
the further step of assuming complete determinism
between brain neurochemical reactions and behavior is
unwarranted. Such a step would necessitate assumptions
that human behavior is completely predictable from the
current brain state. Given the number of neurons involved,
determinism in such a sense, even if “real,” would be
impossible to prove or disprove. Second, we do not really
know what is the neural reality of stochasticity in behavioral
probabilistic models (quantum or classical). At the neuronal
level, it may be the case that such stochasticity is resolved in
a deterministic way. Interestingly, Shanteau & Weiss criticize quantum models in exactly the opposite way, that is,
by arguing that they are too deterministic for modeling
unpredictable human behavior. We are unconvinced by
such arguments. Whichever perspective one adopts in
relation to the determinism of brain neurophysiology, at
the behavioral level there is an obvious need to represent
uncertainty in human life. Therefore, at that level, it
makes sense to adopt modeling frameworks that formalize
uncertainty (whether classical or quantum).
A related issue is whether quantum cognitive models
require quantum processes at the neuronal level (and,
therefore, a quantum brain). This does not appear to be
the case. Two commentaries (Atmanspacher; Blutner
& beim Graben) discuss how incompatibility, a key, if
not the fundamental, property of quantum cognitive
models, can arise in a classical way. This can happen if
measurement is coarse enough that epistemically equivalent states cannot be distinguished. Hameroff, however,
strongly argues in favor of a quantum brain as the source
of quantum behavior (as he says, if it walks and thinks
like a duck then it is a duck). In the past, his work has
mainly addressed the problem of consciousness. But in
this comment, he advances his ideas by providing a detailed
discussion for how his orchestrated objective reduction
(Orch OR) model can be extended to incorporate subspaces and projections for complex thoughts, as would be
required in quantum cognitive models. This is a detailed
proposal for how quantum computations can be directly
implemented in a way that the underlying neuronal processes are quantum mechanical. Such ideas are intriguing,
and he mentions a growing literature on quantum biology,
but currently this direction remains very controversial.
Finally, Stewart & Eliasmith note how the neural
engineering framework can be employed in relation to
vector representations and operations. An important
point is that the problem of neural implementation of
quantum models is not just one of representing vectors
and standard vector operations, such as dot products.
One also needs to consider how characteristics uniquely
relevant to QP theory could be implemented, notably
superposition and incompatibility. It is these latter properties that present the greatest challenge. In other words, for
example, one would need to consider what in the nature of
neural representation gives rise to incompatibility between
different properties (as opposed to merely implementing
subspaces with basis vectors at oblique angles). Stewart &
Eliasmith also note that allowing tensor product operations
in a behavioral model raises questions in relation to how an
increasing number of dimensions are represented, but, for
example, could one not envisage that there are different
sets of neurons to represent the information in relation to
the Happy and Employed questions? Then, when a joint
representation is created, the two sets of neurons are
simply brought together. It is also worth noting that Smolensky et al. (in press) have argued against the perception
that increases in dimensionality caused by tensor products
are problematic. It is currently unclear whether such
dimensionality increases present a problem in relation to
neural implementation.
R6. Rationality
The question of whether an account of human rationality
(or not) should emerge from quantum cognitive models
partly relates to their intended explanatory level.
Navarro & Fuss argue that quantum theory does not
make sense as a normative account of everyday inference.
Instead, their view is that CP theory is the better approach
for computational level analysis, as “it is the right tool for
the job of defining normative inferences in everyday data
analysis.” However, the latter statement is a belief, not an
empirically established truth. Many complex systems are
sensitive to measurement, and a difficulty arises for
classic probability theory when measurement changes the
system being measured. Suppose we obtain measurements
of a variable B under two different conditions: one in which
we measure only B and another in which we measure A
before B. If the prior measurement of A affects the
system, then we need to construct two different CP
models for B, one when A is measured and one when A
is not measured. Unfortunately, these two CP distributions
are stochastically unrelated, which means that the two
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Response/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
distributions have no meaningful relation between each
other (cf. Dzhafarov & Kujala) and a more general
theory is required that can construct both distributions
from a single set of elements. This is what quantum probability theory is designed to do, although as Gelman &
Betancourt point out, there may be other ways to
achieve this generalization.
A related point is that, if for the moment one does away
with normative considerations, both CP and QP theory
have analogous structure and objectives with respect to
cognitive explanation. Both theories make sense as topdown (axiomatic) theories: in neither case are there
assumptions, for example, that the cognitive system represents uncertainty in particular ways. Rather, the claim is
that behavior, however it is produced, is consistent with
the constraints of CP (or QP) theory. Therefore, we think
that a characterization as “top-down” is a more accurate
way to label both CP and QP explanations, without reference to normative considerations. As noted in response to
other commentators (Love; Marewski & Hoffrage), we
aspire to extend quantum models in a way such that
process assumptions are incorporated, and we think that
quantum models offer promise in this respect, but such
extensions start from the “explanatory top.”
Oaksford expresses several intuitions regarding the
rational status of classical probability theory. For example,
he notes that “If one follows the laws of probability in the
Kolmogorov axioms, one will never make a bet one is
bound to lose, and therefore it can provide a rational standard,” but as we argued in the target article, an appeal to
the Dutch book theorem cannot be used to motivate the
rational status of CP theory, as this is only valid with the
psychologically unrealistic assumption of utility as value
maximization. He also notes that “If one follows the laws
of logic, one will never fall into contradiction, and, therefore, logic can provide a rational standard,” but this argument is undermined by Goedel’s incompleteness
theorems, because for many axiomatic systems either not
all conclusions can be axiomatically derived or, if they
can, they will be inconsistent.
Oaksford also observes that classical normative theory is
based on clear intuition. For example, according to standard logic and CP theory, it is irrational to believe both p
and not-p are true. If two events, A versus B, are mutually
exclusive, then this is true for QP theory as well and the two
events have to be compatible in the quantum sense. Also,
the same event B cannot be both true and not true simultaneously. The subtle issue arises when events A versus B
are incompatible, because in this case, knowing A is true
prevents us from concluding that B is true or B is not
true. If A is incompatible with B, then we cannot discuss
their truth values at the same time. Suppose that we
model the truth and falsity of a logical statement as incompatible events, although clearly related in a specific way
(one could employ suitably opposite probability distributions). Then, the events “statement A is true” and “statement A is false” no longer correspond to orthogonal
subspaces, and, therefore, it is no longer the case that
Prob(statement A is true and then statement A is false)
has to be 0. By contrast, Blutner et al. (in press) showed
that the same approach could never produce Prob (statement A is true and statement A is false) ≠ 0 in a corresponding CP model. This is an important point, because
empirically naïve observers do sometimes assign non-zero
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probability values to such statements. In a now wellknown demonstration, Alxatib and Pelletier (2011)
showed that, in the case of describing a person as tall or
not, borderline cases admit the characterization ‘tall and
not tall’ (other predicates and examples were used).
Vague instances with respect to a property p can often be
described as “p and not p” (Blutner et al. in press).
Grace & Kemp observe that the behavior of nonhumans
is sometimes more consistent with classical optimality, than
is that of humans. For example, they note that nonhumans
(e.g., pigeons) often avoid the base rate fallacy. Houston &
Wiesner also suggest that apparent violations of classical
optimality in nonhumans can sometimes be explained if
one revises assumptions about the environment. Does the
debate between classical, quantum rationality apply to
humans as well as nonhumans? The nonhuman studies
typically involve extensive training, and this may provide
the opportunity to form a compatible representation of
the relevant events (by experiencing their joint occurrences). Even with humans, the base rate fallacy can be
greatly reduced, when knowledge about the events is
derived directly from experience. Another possibility is
that perhaps the ability to reason with incompatible representations requires a greater degree of cognitive flexibility (lacking in nonhumans?), so that the same outcome
can be approached from different perspectives.
A related perspective is offered by Ross & Ladyman, who
suggest that Bayesian reasoning may work well in small
worlds, but not in large worlds. We agree that this seems
plausible, in that there would be limited sets of questions
that can be represented in a compatible way (and that, therefore, CP theory can apply), but these sets would be incompatible with other sets, so that when considering the totality of
questions, the requirement from the principle of unicity need
not be fulfilled, and a non-classical picture emerges.
Oaksford is concerned that we do not provide a view of
human rationality, according to QP theory. We suggest a
notion of human rationality in terms of optimality that
arises from formal, probabilistic inference. What marks a
rational agent is the consistency that arises from assessing
uncertainty in the world in a lawful, principled way. Such
a minimalist approach to rationality can be justified partly
by noting that traditional arguments for Bayesian rationality, such as long-term convergence and the Dutch book
theorem, break down under scrutiny. If one accepts this
minimalist approach to rationality, then the question
becomes, which formal system is most suitable for human
reasoning? Oaksford and Chater (2009) themselves convincingly argued against classical logic, because its property of
monotonicity make it unrealistically stringent in relation to
the ever-changing circumstances of everyday life. In an
analogous way, we argue against CP theory, in relation to
the unicity principle. But can we reconcile the perspective
dependence of probabilistic assessment in quantum theory
with the need for predictive accuracy in human decision
making? In its domain of original application (cf. Oaksford;
quantum theory was not created to apply to the macroscopic physical world), in the microscopic world,
quantum prediction can be extremely accurate, despite
all this (seeming) arbitrariness, which arises from perspective dependence. Our point is that perspective dependence
does not reduce accuracy. Is the mental world like the
microscopic physical world, for which quantum theory
was originally developed? To the extent that psychological
References/Pothos & Busemeyer: Can quantum probability provide a new direction for cognitive modeling?
behavior exhibits properties such as incompatibility, interference, and entanglement, we believe that the answer is yes.
Lee & Vanpaemel present another objection to
quantum theory. They note the extent to which a limited
number of physicists have objections to quantum theory.
They provide a telling quote from Jaynes, in which he
strongly questions the value of quantum theory in
physics. (However, we recommend reading Bub [1999]
rather than Jaynes, for a more comprehensive interpretation of quantum theory.) To clarify this issue, physicists
do not object to the formal (mathematical) form of
quantum theory. They debate its interpretation. Our applications to cognition have used the mathematics, and we
have avoided taking any stand on the interpretation of
quantum theory. Leaving aside the fact that no other physical theory has had such a profound impact in changing our
lives (e.g., through the development of the semiconductor
and the laser), few if any physicists think that quantum
theory is going into retirement soon. For completeness, it
is worth noting that Aspect’s work famously and definitively
supported quantum theory against Einstein’s classical
interpretation of Bell’s hypothetical experiment (e.g.,
Aspect et al. 1981). Any introductory quantum mechanics
text will outline the main ideas (e.g., see Isham 1989).
Quantum theory is a formal theory of probability: it
remains one of the most successful in physics and we
wish to explore its possible utility in other areas of human
endeavor.
In conclusion, the wide variety of thought-provoking
comments, ranging across criticisms to empirical challenges
to debates about fundamental aspects of cognition, attest to
Sloman’s view that “quantum theory captures deep insights
about the workings of the mind” (this is part of his review
for Busemeyer & Bruza’s 2012 book, Busemeyer &
Bruza 2012).
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